Engineering functions – D to IMC

1. How to use the DEC2BIN function

The DEC2BIN function converts a decimal number to a binary number.

What is a decimal number?
The decimal system is a positional numeral system that uses 10 as the base, it requires 10 different numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The dot or the decimal point represents decimal fractions which are not whole numbers.

The decimal number 520 has three positions, each with a different weight. It starts with 10^0 on the right and increases by one power on each additional position to the left.

520 = (5*10^2)+(2*10^1)+(0*10^0)

520 = 500 + 20 + 0

What is a binary number?
The binary system is a positional numeral system that uses only two digits: 0 and 1. The binary system is important in our society, many devices like computers, digital cameras, mobile phones and modern cars use binary code to store, process and communicate data. The binary numeral system makes it easy to store and transmit data using binary digits or bits.

Decimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111
16	10000

 

1. DEC2BIN function Syntax

DEC2BIN(number, [places])

2. DEC2BIN function Arguments

number	Required. The decimal integer you want to convert. The sign bit is the most significant bit of number, the following 9 bits are magnitude bits. Negative numbers are represented using two's-complement notation. Maximum binary values are 10 characters.
[places]	Optional. The number of characters to use. If not entered the function uses the number of characters needed to complete the task. This argument allows you to add leading 0s (zeros).

What is a sign bit?

The sign bit indicates whether a binary number is positive or negative, if the bit is 0 the number is positive, if the bit is 1, the number is negative.

What is a magnitude bit?

The remaining 9 bits are magnitude bits which represents the absolute value of the number. An absolute number is a number without the sign.

What is two's-complement notation?

Two’s-complement notation is used to represent negative numbers, the magnitude bits are changed from 0 to 1 and 1 to 0 and adding 1 to the result. For example:

Decimal number +9 using 10 bits is 0000001001.
Decimal number -9 using 10 bits is 1111110110 + 1 = 1111110111

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3. DEC2BIN function example

The image above shows the DEC2BIN function in cells D3 an D4. It calculates the binary values based on the corresponding cells which contain different decimal numbers.

Formula in cell D3:

The first cell B3 contains 9 and cell C3 contains the places number, the DEC2BIN function returns 00001001 in cell D3, the second cell B4 contains 2 and the DEC2BIN function returns 00000010 in cell D4.

The next section describes how these values are calculated in detail.

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4. How is the DEC2BIN function calculated in detail?

Follow these steps to convert from decimal to binary:

1. Divide the decimal number by 2 and write down the quotient and the remainder.
2. Repeat the process with the quotient until it is zero. Make sure to note each quotient and remainder as you calculate each number.
3. The binary number is the remainders from bottom to top.

For example, let us convert decimal number 199 to binary.

199/2 = 99 and remainder 1
99/2 = 49 and remainder 1
49/2 = 24 and remainder 1
24/2 = 12 and remainder 0
12/2 = 6 and remainder 0
6/2 = 3 and remainder 0
3/2 = 1 and remainder 1
1/2 = 0 and remainder 1

The binary number is 11000111 counting from bottom to top.

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5. DEC2BIN Function error

DEC2BIN returns the #NUM! error value if

1. Less than -512 or larger than 511. See cells B3 and C3 in the image above.
2. [places] is negative. See the formula next to cell C5.
3. it requires more than the specified places characters. See the formula next to cell C4.

DEC2BIN returns the #VALUE! error if

1. places is not a number. See the formula next to cell C7.
2. The number is not a valid base 10 number. See cells B4 and C4 in the image above.

The second argument [places]is truncated if it is not an integer. For example, [places] is 4.1111111111 and the DEC2BIN function truncates it to 4. See the formula next to cell C8.

DEC2BIN ignores places and returns a 10-character binary number if the decimal number is negative.

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6. How to convert large decimal numbers to binary?

The following formula converts numbers larger than 511 to their binary representation, it works only for positive decimal numbers. This is a workaround for the DEC2BIN function which has a limit of decimal numbers larger than 511.

Formula in cell  C5:

This formula converts the number in cell B5 to its binary representation and returns it as a text string. Cell B5 contains 512 and the result is 1000000000

The BASE function converts a number into a text representation with a given radix (base).

Function syntax: BASE(number, radix, [min_length])

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DEC2BIN function - Microsoft
Decimal to Binary converter
Decimal to Binary

2. How to use the DEC2HEX function

The DEC2HEX function converts a decimal number to a hexadecimal number.

What is a decimal number?

The decimal system is a positional numeral system that uses 10 as the base, it requires 10 different numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The dot or the decimal point represents decimal fractions which are not whole numbers.
The decimal number 520 has three positions, each with a different weight. It starts with 10^0 on the right and increases by one power on each additional position to the left.

520 = (5*10^2)+(2*10^1)+(0*10^0)

520 = 500 + 20 + 0

What is a hexadecimal number?

A hexadecimal number is a number with a base of 16, for example, the decimal system uses a base of 10. This means that each digit in a hexadecimal number can have 16 possible values, from 0 to 15, however, the letters A to F are used from 10 to 15. See the hexadecimal column in the table below.

Hexadecimal numbers are often used in computers, the reason is they represent four binary digits (bits) with one hexadecimal digit. For example, the binary number 1010 is equivalent to the hexadecimal number A.

Hexadecimals make it easier to write big numbers with less digits, in other words, hexadecimals shorten binary digits considerably. For example, we can use hexadecimal to show the values of colors and MAC addresses in computers.

The following table shows the binary, decimal and hexadecimal values from 0 (zero) to 17.

Decimal	Hexadecimal
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	8
9	9
10	A
11	B
12	C
13	D
14	E
15	F
16	10
17	11

1. DEC2HEX function Syntax

DEC2HEX(number, [places])

2. DEC2HEX function Arguments

number	Required. The decimal integer you want to convert.
[places]	Optional. The number of characters to use. If not entered the function uses the number of characters needed to complete the task. This argument allows you to add leading 0s (zeros).

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3. DEC2HEX function Example

The image above shows the DEC2HEX function in cells D3 an D4. It calculates the hexadecimal values based on the corresponding cells which contain different decimal numbers and [places].

Formula in cell D3:

The first cell B3 contains 9 and cell C3 contains the places number, the DEC2HEX function returns 00000009 in cell D3, the second cell B4 contains 15 and the DEC2HEX function returns 0000000F in cell D4.

The next section describes how these values are calculated in detail.

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4. How is the DEC2HEX function calculated in detail?

There are different methods converting between decimal and hexadecimal, here is one way. Follow these steps to convert from decimal to hexadecimal:

1. Divide the decimal number by 16 and write down the quotient and the remainder.
2. Repeat the process with the quotient until it is zero. Make sure to note each quotient and remainder as you calculate each number.
3. The hexadecimal number is the remainders from bottom to top.

For example, let us convert decimal number 199 to hexadecimal.

199/16 = 12 and remainder 7
12/16 = 0 and remainder 12 (B)

Decimal value 12 is hexadecimal value "B", see the conversion table above. The hexadecimal number is formed from the remainders counting from bottom to top. 199 = B7

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5. DEC2HEX function not working

DEC2HEX returns the #NUM! error value if

1. Number is less than -549,755,813,888 or larger than 549,755,813,887. See cells B3 and C3 in the image above.
2. [places] is negative. See the formula next to cell C5.
3. it requires more than the specified places characters. See the formula next to cell C6.

DEC2HEX returns the #VALUE! error if

1. [places] is not a number. See the formula next to cell C7.
2. The number is not a valid base 10 number. See cells B4 and C4 in the image above.

The second argument [places]is truncated if it is not an integer. For example, [places] is 4.1111111111 and the DEC2HEX function truncates it to 4. See the formula next to cell C8.

DEC2HEX ignores places and returns a 10-character hexadecimal number if the decimal number is negative.

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Useful resources

DEC2HEX function - Microsoft
Decimal to Hexadecimal converter
How to convert decimal to hexadecimal

3. How to use the DEC2OCT function

The DEC2OCT function converts a decimal number to a octal number.

What is a decimal number?

The decimal system is a positional numeral system that uses 10 as the base, it requires 10 different numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The dot or the decimal point represents decimal fractions which are not whole numbers.
The decimal number 520 has three positions, each with a different weight. It starts with 10^0 on the right and increases by one power on each additional position to the left.

520 = (5*10^2)+(2*10^1)+(0*10^0)

520 = 500 + 20 + 0

What is a octal number?

The octal system is a number system with a base of 8 that uses the digits 0, 1, 2, 3, 4, 5, 6 and 7. The octal system is often used in electronics because it is easy to perform a conversion between octal and binary numbers.

Decimal	Octal
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	10
9	11
10	12
11	13
12	14
13	15
14	16
15	17
16	20
17	21

1. DEC2OCT function Syntax

DEC2OCT(number, [places])

2. DEC2OCT function Arguments

number	Required. The decimal integer you want to convert.
[places]	Optional. The number of characters to use. If not entered the function uses the number of characters needed to complete the task. This argument allows you to add leading 0s (zeros).

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3. DEC2OCT function Example

The image above shows the DEC2OCT function in cells D3 an D4. It calculates the octal values based on the corresponding cells which contain different decimal numbers and [places].

Formula in cell D3:

The first cell B3 contains 8 and cell C3 contains the places number, the DEC2OCT function returns 00000010 in cell D3, the second cell B4 contains 16 and the DEC2OCT function returns 00000020 in cell D4.

The next section describes how these values are calculated in detail.

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4. How is the DEC2OCT function calculated in detail?

There are different methods converting between decimal and hexadecimal, here is one way. Follow these steps to convert from decimal to hexadecimal:

1. Divide the decimal number by 8 and write down the quotient and the remainder.
2. Repeat the process with the quotient until it is zero. Make sure to note each quotient and remainder as you calculate each number.
3. The octal number is the remainders from bottom to top.

For example, let us convert decimal number 199 to octal.

199/8 = 24 and remainder 7
24/8 = 3 and remainder 0
3/8 = 0 and remainder 3

The octal number is formed from the remainders counting from bottom to top. 199 = 307

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5. DEC2OCT function not working

DEC2OCT returns the #NUM! error value if

• Number is less than -549,755,813,888 or larger than 549,755,813,887. See cells B3 and C3 in the image above.
• [places] is negative. See the formula next to cell C5.
• it requires more than the specified places characters. See the formula next to cell C6.

DEC2OCT returns the #VALUE! error if

• [places] is not a number. See the formula next to cell C7.
• The number is not a valid base 10 number. See cells B4 and C4 in the image above.

The second argument [places]is truncated if it is not an integer. For example, [places] is 4.1111111111 and the DEC2OCT function truncates it to 4. See the formula next to cell C8.

DEC2OCT ignores places and returns a 10-character hexadecimal number if the decimal number is negative.

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Useful resources

DEC2OCT function - Microsoft
Decimal to Octal Converter
Decimal to Octal - Cuemath

4. How to use the DELTA function

The DELTA function evaluates whether two numerical values are equal. This function is also known as the Kronecker Delta function.

What is the Kronecker Delta?

The Kronecker Delta is a mathematical function that takes two variables, usually positive whole numbers, and returns 1 if they are equal and 0 if not equal.

δ_ij where i and j are variables. i = j returns 1, i <> j returns 0

The Kronecker delta is often used in mathematics, physics, engineering and computer science.

1. DELTA Function Syntax

DELTA(number1, [number2])

2. DELTA Function Arguments

Argument	Text
number1	Required.
[number2]	Optional, default value is 0 (zero).

3. DELTA Function Example

The image above shows arguments number1 in column B, arguments number2 in column C and the result of the DELTA function in column D.

Formula in cell D3:

Row 3 contains numbers 2 and 3, DELTA function  in cell D3 returns 0 (zero). 2 and 3 are not equal. Row 4 has 7 and 7, the DELTA function returns 1. 7 and 7 are equal.

Row 8 shows that negative numbers, in this case, -3 and -3 returns 1. They are equal.

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4. DELTA Function not working

The DELTA function returns a #VALUE error value if number1 or [number2] is nonnumeric. The image above shows the DELTA function returning a #VALUE error in cell D3, the second argument is "A" and is non numeric which is not allowed.

The DELTA function in cell D4 also returns a #VALUE error, it has a boolean value in the first argument. You can convert boolean values to their numerical equivalents by multiplying the boolean value by 1.

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5. DELTA function compared to other comparison functions/operators

The DELTA function compares only numerical values and the output is 1 or 0 (zero). It can't compare an array of values, not text values or boolean values. It is quite limited if you consider array operations.

The equal sign = is a logical operator, it is able to compare numbers, text and boolean values. It also compares a value to an array of values, or an array of values to another array of values. The downside is that it doesn't differentiate between upper and lower letters.

The output from the equal sign is a boolean value TRUE or FALSE, however, it is easy to convert the output to their numerical equivalents TRUE = 1 and FALSE = 0 (zero) by multiplying the output with 1. You can also add a zero if you prefer. It is also possible to use double negations.

The EXACT function is primarily used to compare text strings also considering upper and lower letters. However, it also works with numbers and boolean values as well. It works fine with arrays, the output is either TRUE or FALSE just like the equal sign.

The COUNTIF function is like the equal sign but on steroids. It works fine with arrays and is easy to work with, it can also check if values are smaller/larger than other values. The COUNTIF function returns the actual count meaning a whole number equal or larger than 0 (zero).

The COUNTIFS function is a more advanced than the COUNTIF function.

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Useful resources

DELTA function - Microsoft
Kronecker Delta - Wolfram
Kronecker Delta - Wikipedia

5. How to use the HEX2BIN function

The HEX2BIN function converts a hexadecimal number to a binary number.

What is a hexadecimal number?

A hexadecimal number is a numeral system with a base of 16, for example, the decimal system uses a base of 10. This means that each digit in a hexadecimal number can have 16 possible values, from 0 to 15, however, the letters A to F are used from 10 to 15. See the hexadecimal column in the table below.

Hexadecimal numbers are often used in computers, the reason is they represent four binary digits (bits) with one hexadecimal digit. For example, the binary number 1010 is equivalent to the hexadecimal number A.

Hexadecimals make it easier to write big numbers with less digits, in other words, hexadecimals shorten binary digits considerably. For example, we can use hexadecimal to show the values of colors and MAC addresses in computers.

What is a binary number?

The binary system is a positional numeral system that uses only two digits: 0 and 1. The binary system is important in our society, many devices like computers, digital cameras, mobile phones and modern cars use binary code to store, process and communicate data. The binary numeral system makes it easy to store and transmit data using binary digits or bits.

Hexadecimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
A	1010
B	1011
C	1100
D	1101
E	1110
F	1111
10	10000

1. HEX2BIN Function Syntax

HEX2BIN(number, [Places])

2. HEX2BIN Arguments

number	Required. The hexadecimal number you want to convert to a decimal number.
[Places]	Optional. How many digits to use, if omitted HEX2BIN uses the minimum number of required digits. Tip! Use [Places] for padding with leading 0's (zeros). HEX2BIN function ignores [Places] argument if the hexadecimal is negative.

3. HEX2BIN Function Example

The image above demonstrates the HEX2BIN function cell C3, C4, C5, and C6. It calculates the binary values based on the corresponding cells which contain different hexadecimal numbers.

Formula in cell C3:

The first cell B3 contains 1 and  the HEX2BIN function returns 1 in cell D3, the second cell B4 contains F and the HEX2BIN function returns 1111 in cell D4.

4. How is hexadecimal to binary calculated?

Binary	0	1	10	11	100	101	110	111	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Convert every hexadecimal to a binary using the table above.

4.1 Example 1

Hexadecimal value 664 is calculated to a binary like this:

Third hexadecimal value: 6 = 0110

Second hexadecimal value: 6 = 0110

First hexadecimal value: 4 = 0100

equals

0110 0110 0100

4.2 Example 2

Hexadecimal value F9F is calculated to binary like this:

F = 1111

9 = 1001

F = 1111

equals

1111 1001 1111

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5. HEX2BIN Function not working

HEX2BIN returns the #NUM! error value if

• Number is less than FFFFFFFE00 or larger than 1FF. See cells B3 and C3 in the image above.
• [places] is negative. See the formula next to cell C5.
• it requires more than the specified places characters. See the formula next to cell C6.
• The number is not a valid base 16 number. See cells B4 and C4 in the image above.

HEX2BIN returns the #VALUE! error if

• [places] is not a number. See the formula next to cell C7.

The second argument [places]is truncated if it is not an integer. For example, [places] is 4.1111111111 and the HEX2BIN function truncates it to 4. See the formula next to cell C8.

HEX2BIN ignores places and returns a 10-character binary number if the hexadecimal number is negative.

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Useful links

HEX2BIN function - Microsoft
Hex to Binary converter
Hexadecimal - Wikipedia

6. How to use the HEX2DEC function

The HEX2DEC function converts a hexadecimal number to a decimal number.

What is a hexadecimal number?

A hexadecimal number is a numeral system with a base of 16, for example, the decimal system uses a base of 10. This means that each digit in a hexadecimal number can have 16 possible values, from 0 to 15, however, the letters A to F are used from 10 to 15. See the hexadecimal column in the table below.

Hexadecimal numbers are often used in computers, the reason is they represent four binary digits (bits) with one hexadecimal digit. For example, the binary number 1010 is equivalent to the hexadecimal number A.

Hexadecimals make it easier to write big numbers with less digits, in other words, hexadecimals shorten binary digits considerably. For example, we can use hexadecimal to show the values of colors and MAC addresses in computers.

What is a decimal number?
The decimal system is a positional numeral system that uses 10 as the base, it requires 10 different numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The dot or the decimal point represents decimal fractions which are not whole numbers.

The decimal number 520 has three positions, each with a different weight. It starts with 10^0 on the right and increases by one power on each additional position to the left.

520 = (5*10^2)+(2*10^1)+(0*10^0)

520 = 500 + 20 + 0

1. HEX2DEC Function Syntax

HEX2DEC(number)

2. HEX2DEC Arguments

number

Required. The hexadecimal number you want to convert to a decimal number. The sign bit is the most significant bit, the remaining 39 bits are magnitude bits. Negative numbers are represented using two's-complement notation.

What is a sign bit?

The sign bit indicates whether the hexadecimal number is positive or negative, if the bit is 0 the number is positive, if the bit is 1, the number is negative.

What is a magnitude bit?

The remaining 39 bits are magnitude bits which represents the absolute value of the number. An absolute number is a number without the sign.

What is two's-complement notation?

Two’s-complement notation is used to represent negative numbers, the magnitude bits are changed from 0 to 1 and 1 to 0 and adding 1 to the result. For example:

Hexadecimal number 7FFFFFFFFF using 10 bits is 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 which is a positive decimal number.
Hexadecimal number 8FFFFFFFFF using 10 bits is 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 which is a negative decimal number.

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3. HEX2DEC Function Example

The image above demonstrates the HEX2DEC function cell C3, C4, C5, and C6. It calculates the decimal values based on the corresponding cells which contain different hexadecimal numbers.

Formula in cell D3:

The first cell B3 contains 1 and the HEX2DEC function returns 1 in cell C3, the second cell B4 contains F and the HEX2DEC function returns 15 in cell C4.

The third cell B5 contains FFFFFFFFFF and the HEX2DEC function returns -1 in cell C5, the fourth cell B6 contains FFFFFFFFFE and the HEX2DEC function returns -2 in cell C6.

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4. How is hexadecimal to decimal calculated?

Hexadecimal	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Decimal	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

xyz

becomes

x * (16 ^ 2)

y * (16 ^ 1)

z * (16 ^ 0)

equals

x * (16 ^ 2) + y * (16 ^ 1)  + z * (16 ^ 0)

4.1 Example 1

Hexadecimal value 664 is calculated to decimal like this:

Third hexadecimal value: 6*16^2 = 6*256 = 1536

Second hexadecimal value: 6*16^1 = 6*16 = 96

First hexadecimal value: 4*16^0 = 4*1 = 4

1536+96+4

equals

1636.

4.2 Example 2

Hexadecimal value F9F is calculated to decimal like this:

F = 15, see the table above. F is the third character from the right. 15*16^2 = 3840

9 is the second character from the right. 9*16^1=144

F = 15

Add the numbers: 3840 + 144 + 15 equals 3999

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External links

Hexadecimal to decimal
Hexadecimal to Decimal converter
Hexadecimal to Decimal - Cuemath

7. How to use the HEX2OCT function

The HEX2OCT function converts a hexadecimal number to an octal number.

What is a hexadecimal number?

A hexadecimal number is a numeral system with a base of 16, for example, the decimal system uses a base of 10. This means that each digit in a hexadecimal number can have 16 possible values, from 0 to 15, however, the letters A to F are used from 10 to 15. See the hexadecimal column in the table below.

Hexadecimal numbers are often used in computers, the reason is they represent four binary digits (bits) with one hexadecimal digit. For example, the binary number 1010 is equivalent to the hexadecimal number A.

Hexadecimals make it easier to write big numbers with less digits, in other words, hexadecimals shorten binary digits considerably. For example, we can use hexadecimal to show the values of colors and MAC addresses in computers.

What is a octal number?

The octal system is a numeral system with a base of 8 that uses the digits 0, 1, 2, 3, 4, 5, 6 and 7. The octal system is often used in electronics because it is easy to perform a conversion between octal and binary numbers.

The following table shows the hexadecimal, octal and decimal values from 0 (zero) to 17.

Hexadecimal	Octal	Decimal
0	0	0
1	1	1
2	2	2
3	3	3
4	4	4
5	5	5
6	6	6
7	7	7
8	10	8
9	11	9
A	12	10
B	13	11
C	14	12
D	15	13
E	16	14
F	17	15
10	20	16
11	21	17

1. HEX2OCT Function Syntax

HEX2OCT(number, [places])

2. HEX2OCT Function Arguments

number	Required. The hexadecimal number you want to convert to an octal number. The sign bit is the most significant bit. The remaining 39 bits are magnitude bits. Negative numbers are represented using two's-complement notation.
[places]	Optional. The number of digits to show, in other words, this creates leading zeros if not all digits are used.

What is a sign bit?

The sign bit indicates whether the hexadecimal number is positive or negative, if the bit is 0 the number is positive, if the bit is 1, the number is negative.

What is a magnitude bit?

The remaining 39 bits are magnitude bits which represents the absolute value of the number. An absolute number is a number without the sign.

What is two's-complement notation?

Two’s-complement notation is used to represent negative numbers, the magnitude bits are changed from 0 to 1 and 1 to 0 and adding 1 to the result. For example:

Hexadecimal number 7FFFFFFFFF using 10 bits is 0111 1111 1111 1111 1111 1111 1111 1111 1111 1111 which is a positive decimal number.
Hexadecimal number 8FFFFFFFFF using 10 bits is 1000 1111 1111 1111 1111 1111 1111 1111 1111 1111 which is a negative decimal number.

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3. HEX2OCT Function example

The image above shows the HEX2OCT function in cell C3, it calculates the octal number based on the specified number in cell B3.

Formula in cell C3:

The first example shows the HEX2OCT function in cell C3, the argument is specified in cell B3 which is 1 in hexadecimal and it returns 1 in octal.

Cell B4 contains F in hexadecimal which the HEX2OCT function converts to 17 in the octal numeral system shown in cell C4.

The third example converts "FFFFFFFFFF" to octal, the result is shown in cell C5 which is 7777777777 in octal.

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4. How is the HEX2OCT Function calculated?

There is no easy way to convert from hexadecimal to octal by manually calculating the values. You can convert from hexadecimal to 4 digit binary and then put the binary digits in groups of three. Lastly, convert from three digit binary to octal, however, it is probably easier to just convert from hexadecimal to octal using the following table:

Hexadecimal	Octal
0	0
1	1
2	2
3	3
4	4
5	5
6	6
7	7
8	10
9	11
A	12
B	13
C	14
D	15
E	16
F	17

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5. HEX2OCT Function not working

The hexadecimal argument can't contain more than 10 characters.

HEX2OCT ignores places and returns a 10-character octal number if the hexadecimal is negative.

The smallest hexadecimal value is FFE0000000 and the largest is 1FFFFFFF. Values larger or smaller than the specified range are not allowed.

HEX2OCT returns the #NUM! error value if the hexadecimal value is not valid.

HEX2OCT function returns #NUM! error value if it requires more than the places argument allows.

The places argument is truncated if not an integer.

HEX2OCT returns the #VALUE! error value if the places argument is a non-numeric value.

The HEX2OCT function returns #NUM! error if places argument is negative.

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Useful links

HEX2OCT function - Microsoft
Hex to Octal converter
Hexadecimal Number System - Cuemath

8. How to use the IMABS function

The IMABS function calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.

A complex number consists of an imaginary number and a real number, complex numbers let you solve polynomial equations using imaginary numbers if no solution is found with real numbers. It has applications in engineering such as electronics, electromagnetism, signal analysis, and more.

1. IMABS Function Syntax

IMABS(inumber)

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2. IMABS Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMABS function example

The IMABS function calculates the modulus of a complex number, IMABS probably stands for imaginary absolute. The absolute value is the same as the modulus.

The modulus of a complex number is the distance of the complex number from the origin in the complex plane. It is the square root of the sum of the squares of the real part and the imaginary part of the complex number. If the complex number is Z then the modulus is denoted |Z|.

The image above shows a chart of the complex plane, complex number 9+12i is the blue line ending with an arrow. The complex plane has an imaginary axis and a real axis, the dashed circle shows the modulus value when it crosses both the imaginary axis and the real axis.

Formula:

The formula calculates the modulus based on the value in cell C26 which is "9+12i" in this example. The formula returns 15 which the dashed circle also shows when it crosses the imaginary and real axis. Section 4 below explains in greater detail how the IMABS function calculates the modulus.

The modulus is needed when you want to:

• convert complex numbers from rectangular form to polar form or vice versa.
• compare sizes or magnitudes of different complex numbers.
• calculate the distance between two complex numbers

3.1 Explaining formula

Step 1 - Populate arguments

IMABS(inumber)

becomes

IMABS(C26)

Step 2 - Evaluate IMABS function

IMABS(C26)

becomes

IMABS("9+12i")

and returns 15.

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4. How is the modulus calculated in detail?

The formula to calculate the absolute value or modulus z from a complex number is based on the Pythagorean theorem.

z² = x²+y²

The IMABS function calculates the absolute value using this formula which is based on the Pythagorean theorem:

IMABS(z) = |z| =√(x²+y²)

x is the real coefficient and y is the imaginary coefficient of the complex number.

The modulus of a complex number is how far it is from the point where the real and imaginary axes cross (0,0). It is the square root of the real part squared plus the imaginary part squared.

Here is how the modulus is calculated for complex number 9+12i:

z = x + yi

z = 9 + 12i

IMABS(z) = |z| =√(x²+y²)

IMABS(z) = |z| =√(9²+12²)

IMABS(z) = |z| =√(81+144)

IMABS(z) = |z| =√225

IMABS(z) = |z| =15

5. How imaginary numbers were invented

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6. How to convert complex numbers to polar form?

Formula in cell D4:

Complex numbers are usually presented in this form

z = x + yi

or

z = x + yj

However, complex numbers can also be represented in polar form:

z = r*(cos θ + isin θ)

In other words, theta θ in the polar form is calculated using the IMARGUMENT based on complex numbers.

Pythagorean Theorem

r² = x² + y²

To calculate the absolute value we can use this formula:

r = √(x²+y²)

Excel has a function that does this for you, the IMABS function calculates the absolute value based on complex numbers.

Explaining formula

Step 1 - Calculate theta θ

The IMARGUMENT function calculates theta θ which is an angle displayed in radians based on complex numbers in rectangular form.

Function syntax: IMARGUMENT(inumber)

IMARGUMENT(C4)

becomes

IMARGUMENT("12+9i")

and returns

0.643501108793284

Step 2 - Calculate the absolute value

The IMABS function calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.

Function syntax: IMABS(inumber)

IMABS(C4)

becomes

IMABS("12+9i")

and returns

15

Step 3 - Join calculations with text

The ampersand character lets you concatenate values in an Excel Formula.

IMABS(C4)&"(cos "&IMARGUMENT(C4)&" + isin "&IMARGUMENT(C4)&")"

Step 4 - Evaluate the formula

IMABS(C4)&"(cos "&IMARGUMENT(C4)&" + isin "&IMARGUMENT(C4)&")"

becomes

15&"(cos "&0.643501108793284&" + isin "&0.643501108793284&")"

and returns

15(cos 0.643501108793284 + isin 0.643501108793284)

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Useful links

IMABS function - Microsoft

9. How to use the IMAGINARY function

The IMAGINARY function calculates the imaginary number of a complex equation in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

1. IMAGINARY Function Syntax

IMAGINARY(inumber)

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2. IMAGINARY Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMAGINARY function example

A complex number consists of an imaginary number and a real number, complex numbers let you solve polynomial equations using imaginary numbers if no solution is found with real numbers.

A complex number in rectangular form can be described as z = x + yi or z = x + yj text form in Excel. The IMAGINARY function extracts the imaginary value from the complex number.

Formula in cell D3:

The imaginary coefficient is the number ending with a i or j, this number is what the IMAGINARY function extracts from a complex number.

3.1 Explaining formula

Step 1 - Populate arguments

IMAGINARY(inumber)

becomes

IMAGINARY(B25)

Step 2 - Evaluate IMAGINARY function

IMAGINARY(C3)

becomes

IMAGINARY("3+4i")

and returns 4.

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4. When to use the IMAGINARY function?

Use the IMAGINARY function when you want to

• add, subtract, multiply and divide complex numbers.
• calculate the modulus which is the distance from the origin to the point representing the complex number.
• graph complex numbers
• calculate the complex determinant of a 2x2 matrix

The links above points to articles explaining how to manually calculate these properties, however, Excel has functions so you don't need to calculate them manually:

IMSUM | IMSUB | IMPRODUCT | IMDIV | IMARGUMENT | IMABS

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5. How to plot the imaginary part of a complex number on a chart

The chart above shows how to represent the complex number “3+4i” on the complex plane. The complex plane has two axes: the x-axis is for real numbers and the y-axis is for imaginary numbers. The blue line with an arrow points from the origin (0,0) to (3,4), which is the location of “3+4i” on the plane.

The horizontal dashed line marks the imaginary part of the complex number “3+4i” on the y-axis. The IMAGINARY function can extract the real part from any complex number, which is useful for plotting complex numbers on charts.

A complex number has both a real part and an imaginary part, and we need both of them to plot a complex number on the plane.

5.1 Calculate the real and imaginary parts of a complex number

To plot a complex number on the complex plane, we have to find its real and imaginary parts separately. Cell B25 has the complex number in rectangular form.

Formula in cell C25:

The IMREAL function extracts the real number from the complex number in cell B25.

Formula in cell D25:

The IMAGINARY function extracts the imaginary number from the complex number in cell B25.

To plot a line, we need to use coordinates from the origin (0,0), so I have entered 0 (zero) in cells C24 and D24. The scatter chart that we will create soon requires a blank row between the line coordinates to show two separate lines that are not connected.

The dashed line also needs two points on the chart to be displayed correctly. It starts from where the complex number is (3,4) and ends at the y-axis. The line is horizontal, so the end point must have an real part of 0 (zero).

5.2 Insert a scatter chart

The following steps describe how to plot a complex number and the corresponding real number.

1. Select cell range C24:D28.
2. Go to tab "Insert" on the ribbon.
3. Press with left mouse button on the "Insert Scatter (x,y) or Bubble chart" button.
   
4. A popup menu appears, press with left mouse button on the "Scatter with straight lines".

A chart shows up on the worksheet, move the chart to its desired location.

Change the chart so it shows the complex number as a line with an ending arrow, the imaginary number as a dashed line and so on. Here are detailed instructions:
How to plot theta θ - Argand diagram

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Useful links

IMAGINARY function - Microsoft

10. How to use the IMARGUMENT function

The IMARGUMENT function calculates the argument theta θ which is an angle displayed in radians based on complex numbers in rectangular form like z = x + yi or z = x + yj.

1. IMARGUMENT Function Syntax

IMARGUMENT(inumber)

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2. IMARGUMENT Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMARGUMENT function example

The image above shows how to calculate the θ with the IMARGUMENT function based on the corresponding complex numbers on the same row specified in cell B3. The dashed circle represents the complex modulus |z| of "2+3i" and the angle theta represents its complex argument.

Formula in cell C3:

The IMARGUMENT function returns a value expressed in radians, the image above shows a chart where theta θ is displayed in degrees.

3.1 Explaining formula

Step 1 - Populate arguments

IMARGUMENT(inumber)

becomes

IMARGUMENT(B3)

Step 2 - Evaluate IMARGUMENT function

IMARGUMENT(B3)

becomes

IMARGUMENT("2+3i")

and returns

0.982793723247329

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4. How is the theta θ calculated in detail?

The IMARGUMENT function calculates theta θ expressed in radians using this formula:

θ = tan^-1(y/x) for x>0

Here is how 0.982793723247329 is calculated:

tan θ = y/x

becomes

θ = tan^-1(y/x)

tan^-1(y/x)

becomes

tan^-1(3/2)

equals

0.982793723247329

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5. How to convert angle theta θ from radians to degrees

The following formula calculates theta θ based on a given complex number in rectangular form, the result is a value expressed in radians.

The DEGREE function takes the radian value and converts it to degrees.

Formula:

5.1 Explaining formula

Step 1 - Calculate theta θ in radians

The IMARGUMENT function calculates theta θ which is an angle displayed in radians based on complex numbers in rectangular form.

Function syntax: IMARGUMENT(inumber)

IMARGUMENT(B3)

becomes

IMARGUMENT("2+3i")

and returns

0.982793723247329

Step 2 - Convert radians to degrees

The DEGREES function calculates degrees from radians.

Function syntax: DEGREES(angle)

DEGREES(IMARGUMENT(B3))

becomes

DEGREES(0.982793723247329)

and returns

56.3099324740202

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6. How to plot theta θ - Argand diagram

The image above shows an Argand diagram which is a chart of complex numbers, the dashed circle is the modulus of the complex numbers.

C1 = 2  +3i

|C1| = |2  +3i| = Modulus = square root of 13 = 3.605551

theta θ = tan^-1(3/2) = 0.982793723247329 radians = 56.3099324740202 degrees

Change the complex number in cell B31 and the chart adjusts accordingly.

6.1 Calculate the numbers

We need to setup the worksheet before we can insert the scatter chart.

Value in cells C24, D27, C30, and D30: 0

Formula in cell D24, D25, D28, and D31:

Formula in cell C25. C27, C28, and C31:

Value in cell D27: 0

Value in cells C30 and D30: 0

Formula in cell B33:

6.2 Explaining formula in cell B33

The value in cell B33 will be used in the chart, I will show you how to do that below.

Step 1 - Calculate theta θ

The ATAN function calculates the arctangent of a number.

Function syntax: ATAN(number)

ATAN(D31/C31)

Step 2 -  Convert radians to degree

The ATAN function calculates the arctangent of a number.

Function syntax: ATAN(number)

DEGREES(ATAN(D31/C31))

Step 3 - Round degrees to one digit

The ATAN function calculates the arctangent of a number.

Function syntax: ATAN(number)

ROUND(DEGREES(ATAN(D31/C31)),1)

Step 4 - Concatenate values

"θ: "&ROUND(DEGREES(ATAN(D31/C31)),1)&"°"

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6.3 Create values for a circle on the chart

The following formulas in cells C33 and D33 calculates values that creates a circle on the chart, the image above shows a circle on a chart.

Formula in cell C33:

6.4 Explaining formula in cell C33

Step 1 - Calculate each 15 degree segment

The PI function returns the number pi (¶).

Function syntax: PI()

PI()/12

Step 2 - Create a sequence

The ROWS function calculate the number of rows in a cell range.

Function syntax: ROWS(array)

ROWS($A$1:A1)-1

Step 3 - Multiply segment with sequence

The ROWS function calculate the number of rows in a cell range.

Function syntax: ROWS(array)

PI()/12*(ROWS($A$1:A1)-1)

Step 4 - Calculate sine

The SIN function calculates the sine of an angle.

Function syntax: SIN(number)

SIN(PI()/12*(ROWS($A$1:A1)-1))

Step 5 - Calculate modulus

The IMABS function calculates the absolute value (modulus) of a complex number in x + yi or x + yj text format.

Function syntax: IMABS(inumber)

IMABS($B$31)

Step 6 - Multiply sine with modulus

SIN(PI()/12*(ROWS($A$1:A1)-1))*IMABS($B$31)

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Formula in cell D33:

The formula in cell D33 is the same as in cell C33, except it calculates the cosine instead.

The cos function calculates the cosine of an angle.

Function syntax: COS(number)

Copy cells C33 and D33, paste to 24 rows below so the entire circle can be charted.

6.5 Insert "Scatter with smooth lines" chart

1. Select cell range B24:D57.
2. Go to tab "Insert" on the ribbon.
3. Press with left mouse button on the "Scatter (x,y) or Bubble chart" button on the ribbon.
4. A popup menu appears. Press with left mouse button on the "Scatter with smooth lines" button.

A new chart is now created. Move the chart to the desired location.

Select the chart, the boundaries now have "handles" that you can drag with the mouse to resize the chart.

1. Double-press with left mouse button on the circle to select all lines on the chart and open the settings pane.
   
2. Press with left mouse button on the "Fill & Line" button on the settings pane.
3. Press with left mouse button on the "Line" on the settings pane, new settings in context to "Line" shows up.
4. Press with left mouse button on the "Dash type" button, select a dashed line.
   
5. All lines are now dashed. Press with left mouse button on the "Color" button and change to black.

6.6 Change the complex number line on the chart

1. Select only the "complex number" line. Press with left mouse button on the "complex number" line once to select it if you have all lines selected.
   If no lines are selected then press with left mouse button on the "complex number" line twice to select it.
2. Press with left mouse button on the "Line" on the settings pane. Change color, dash type and add an ending arrow.
   
3. Repeat these steps for the imaginary and real lines.

Change the chart tite, use the settings pane to change the axis line widths and colors, remove chart grids.

1. Select the "complex number" line (blue).
2. Press with left mouse button on the "plus" sign next to the chart.
   
3. Press with left mouse button on the check box next to "Data Labels" to enable data labels.
4. Open the settings pane.
5. Go to "Label options"
   
6. Press with left mouse button on checkbox next to "Value From Cells", select cell B33.
   
7. Move the data label to its destination.

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Useful links

IMARGUMENT function - Microsoft
Argument of a complex number
What Is Argument Of Complex Number?

11. How to use the IMCONJUGATE function

The IMCONJUGATE function calculates the complex conjugate of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

1. IMCONJUGATE Function Syntax

IMCONJUGATE(inumber)

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2. IMCONJUGATE Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMCONJUGATE function Example

The image above demonstrates a formula in cell D3 that calculates the complex conjugate of a complex number specified in cell C3. The chart shows the complex number 3+4i on the complex plane, it also shows the complex conjugate of 3+4i which is 3-4i.

Formula in cell D3:

3.1 Explaining formula

Step 1 - Populate arguments

IMCONJUGATE(inumber)

becomes

IMCONJUGATE(C3)

Step 2 - Evaluate IMCONJUGATE function

IMCONJUGATE(C3)

becomes

IMCONJUGATE("3+4i")

and returns

"3-4i".

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4. How is the complex conjugate calculated?

The complex conjugate is calculated by changing the sign of the imaginary value of a complex number. The real part and the imaginary part are equal in magnitude however the imaginary part is opposite in sign.

IMCONJUGATE(x+yi) = z̄ = (x-yi)

The complex conjugate is often denoted as z̄.

Z = x+yi

z̄ = x-yi

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5. How to calculate the product of a complex number and its complex conjugate

The product of a complex number and its conjugate is a real number.

z*z̄ = |z|^2

z*z̄ = (3+4i)*(3-4i) = 9-12i+12i+16 = 25

|z|^2 = 5^2 = 25

Formula in cell D3:

The image above shows a chart that has a complex number, its complex conjugate and the product plotted on a complex plane. The light blue line is the complex number, the green line is its complex conjugate and the orange line is the product of those two complex numbers.

5.1 Explaining formula in cell D3

Step 1 - Calculate the complex conjugate

The IMCONJUGATE function calculates the complex conjugate of a complex number in x + yi or x + yj text format.

Function syntax: IMCONJUGATE(inumber)

IMCONJUGATE(B25)

becomes

IMCONJUGATE("3+4i")

and returns

"3-4i"

Step 2 - Calculate the product of a complex number and its complex conjugate

The IMPRODUCT function calculates the product of complex numbers in x + yi or x + yj text format.

Function syntax: IMPRODUCT(inumber1, [inumber2], ...)

IMPRODUCT(B25,IMCONJUGATE(B25))

becomes

IMPRODUCT("3+4i","3-4i")

and returns 25.

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6. How to calculate the modulus of a complex conjugate

The conjugate of a complex number z̄ has the same modulus as the complex number z.

|z̄| = |z|

The image above shows the modulus as a grey dashed circle on the chart, both the complex number and its complex conjugate has the same modulus.

Formula in cell C24:

IMABS(B24)

Formula in cell B25:

Formula in cell C25:

The formulas above also show that the modulus of a complex number is the same as the modulus of its complex conjugate. Cell C24 contains "3+4i", the complex conjugate is "3-4i" displayed in cell B25. The modulus is 5 for both the complex number and its complex conjugate.

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7. How to calculate the conjugate of a conjugate complex number

The conjugate of a conjugate of a complex number is the original complex number,

Formula in cell D3:

The above formula demonstrates that it is the original complex number. Cell C3 contains the original complex number, cell D3 contains a formula that calculates the conjugate of a conjugate of the complex number in cell C3. The result is the exact same complex number.

Explaining formula

Step 1 - Calculate the complex conjugate

The IMCONJUGATE function calculates the complex conjugate of a complex number in x + yi or x + yj text format.

Function syntax: IMCONJUGATE(inumber)

IMCONJUGATE(C3)

becomes

IMCONJUGATE("3+4i")

and returns

"3-4i"

Step 2 - Calculate the complex conjugate

The IMCONJUGATE function calculates the complex conjugate of a complex number in x + yi or x + yj text format.

Function syntax: IMCONJUGATE(inumber)

IMCONJUGATE(IMCONJUGATE(C3))

becomes

IMCONJUGATE("3-4i")

and returns "3+4i"

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Useful links

IMCONJUGATE function - Microsoft
Complex Conjugate
Complex conjugate - wikipedia

12. How to use the IMCOS function

What is the IMCOS function?

The IMCOS function calculates the cosine of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

What is the cosine?

The cosine is a trigonometric function that relates to an angle θ in a right triangle to the ratio of the length of the side adjacent the angle and the length of the longest side (hypotenuse) of the triangle. A right triangle has one angle that measures 90° or π/2 radians which is approximately 1.5707963267949 radians.

What is the difference between sine and the complex sine?

The difference between cosine and complex cosine is that the former is defined for real numbers only, while the latter is defined for complex numbers as well. The cosine of a complex number has some similarities to the cosine of a real number, such as periodicity.

The cosine function is periodic considering the angle, meaning it repeats its values after a certain interval. The period of the cosine function is 2π or 360 degrees.

cosine z = (e^iz + e^-iz)/2

z - complex number
i - imaginary unit

1. IMCOS Function Syntax

IMCOS(inumber)

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2. IMCOS Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMCOS function example

The image above demonstrates a formula in cell B28 that calculates the cosine of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell B28:

3.1 Explaining formula

Step 1 - Populate arguments

IMCOS(inumber)

becomes

IMCOS(B25)

Step 2 - Evaluate IMCOS function

IMCOS(B25)

becomes

IMCOS("2+2i")

and returns

-1.56562583531574-3.29789483631124i

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4. How is the IMCOS function calculated in detail?

The cosine of a complex number is calculated like this:

C = x + yi

cos(C) = cos(x)*cosh(y) - sin(x)*sinh(y)i

For example, C=2+2i

cos(2+2i) = cos(2) cosh(2) - sin(2) sinh(2)i

becomes

cos(2+2i) = -0.416146836547142*3.76219569108363 + 0.909297426825682*3.62686040784702i

equals

cos(2+2i) = -1.56562583531574 - 3.29789483631124i

sin - calculates the sine of a number
cos - calculates the cosine of a number
cosh - calculates the hyperbolic cosine of a number
sinh - calculates the hyperbolic sine of a number

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5. IMCOS function not working - #NUM error

The IMCOS function returns a #NUM error if the provided argument is not a valid complex number. The image above shows a worksheet with an invalid complex number specified in cell B25: 2+2

Excel needs an i or j in the complex number to work properly, correct the mistake and the IMCOS function will work again.

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Useful links

IMCOS function - Microsoft
The Complex Cosine and Sine Functions - Mathonline
Cosine of Complex Number - Proofwiki

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13. How to use the IMCOSH function

What is the IMCOSH function?

The IMCOSH function calculates the hyperbolic cosine of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

What is the hyperbolic cosine?

Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them.

Trigonometric functions use a circle, while hyperbolic functions use a hyperbola. The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola. The chart below shows a circle containing the trigonometric functions.

What is a hyperbola?

The equation of a hyperbola with a horizontal axis is
(x²/ a²) - (y² / b²) = 1
where a and b are positive constants.

A circle has a constant distance from the center point, while a hyperbola is a curve that has two focus points (+ae, 0), and (-ae, 0).

What is the difference between hyperbolic cosine and complex hyperbolic cosine?

The difference between hyperbolic cosine and hyperbolic cosine for complex numbers is that the former is defined for real numbers, while the latter is defined for complex numbers.

The hyperbolic cosine of a real number x is defined as
cosh(x) = (e^x + e^-x)/2
Natural number e is the base of the natural logarithm.

The complex hyperbolic cosine of a complex number z = x + yi is defined as cosh(z) = cosh(x)cos(y) + i sinh(x)sin(y)
Complex numbers has i as the imaginary unit.

1. IMCOSH Function Syntax

IMCOSH(inumber)

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2. IMCOSH Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMCOSH function example

The image above demonstrates a formula in cell B28 that calculates the hyperbolic cosine of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell B28:

The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.

Complex number 2+i is the light blue line in the first quadrant. The hyperbolic cosine of 2+i is the green line also shown in the first quadrant.

3.1 Explaining formula

Step 1 - Populate arguments

IMCOSH(inumber)

becomes

IMCOSH(B25)

Step 2 - Evaluate the IMCOSH function

IMCOSH(B25)

becomes

IMCOSH("2+i")

and returns

2.03272300701967+3.0518977991518i

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4. How the IMCOSH function is calculated in detail

The hyperbolic cosine of a complex number is calculated like this:

C = x + yi

cosh(x + yi) = cosh(x)*cos(y) + isinh(x)*sin(y)

For example, C=2+i

cosh(2 + i) = cosh(2)*cos(1) + isinh(2)*sin(1)

becomes

cosh(2 + i) = 3.76219569108363*0.54030230586814 + 3.62686040784702*0.841470984807897i

equals

cosh(2 + i) = 2.03272300701967+3.0518977991518i

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5. IMCOSH function not working

The IMCOSH function returns a #NUM error if the provided argument is not a valid complex number.

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Useful links

IMCOSH function - Microsoft
Hyperbolic functions for complex numbers
Hyperbolic functions – Graphs, Properties, and Examples
Hyperbolic curve fitting in Excel

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14. How to use the IMCOT function

The IMCOT function calculates the cotangent of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

What are complex numbers?

What is a cotangent?

The trigonometric cotangent is a function that relates an angle of a right-angled triangle to the ratio of adjacent side and the opposite side. It is also the inverse of the tangent, cot(θ) = 1/tan(θ).

What is the difference between cotangent and complex cotangent?

The difference between cotangent and cotangent for complex numbers is that the former is defined for real numbers, while the latter is defined for complex numbers.

The cotangent of a real number x is defined as cot(x) = i(e^iθ + e^-iθ)/(e^iθ - e^-iθ)

Natural number e is the base of the natural logarithm.

The complex cotangent of a complex number z = x + yi is defined as cot(x + yi) = (cosh(x)*cosh(y) - isin(x)*sinh(y)) / (sin(x)*cosh(y) + icos(x)*sinh(y))
Complex numbers has i as the imaginary unit.

1. IMCOT Function Syntax

IMCOT(inumber)

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2. IMCOT Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMCOT function example

The image above shows a formula in cell B28 that calculates the cotangent of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell B28:

The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.

Complex number 2+i is the light blue line in the first quadrant. The cotangent of 2+i is the green line located in the third quadrant.

3.1 Explaining formula

Step 1 - Populate arguments

IMCOT(inumber)

becomes

IMCOT(B25)

Step 2 - Evaluate the IMCOT function

IMCOT(C3)

becomes

IMCOT("2+i")

and returns

-0.171383612909185-0.821329797493852i

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4. How is the IMCOT function calculated in detail?

The cotangent of a complex number is calculated like this:

cot(x + yi) = (cos(x)*cosh(y) - isin(x)*sinh(y)) / (sin(x)*cosh(y) + icos(x)*sinh(y))

For example, C=2+i

cot(2+i) = (cos(2)*cosh(1) - isin(2)*sinh(1)) / (sin(2)*cosh(1) + icos(2)*sinh(1))

becomes

cot(2+i) = (-0.416146836547142*1.54308063481524 - i0.909297426825682*1.1752011936438) / (0.909297426825682*1.54308063481524 + i-0.416146836547142*1.1752011936438)

becomes

cot(2+i) = (-0.64214812471552 - i1.06860742138278) / (1.40311925062204 + i-0.489056259041294)

equals

-0.171383612909184-0.821329797493853i

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5. IMCOT function not working - #NUM error

The IMCOT function returns a #NUM error if the provided argument is not a valid complex number.

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Useful links

IMCOT function - Microsoft
Cotangent of Complex Number
How to find cotangent of complex numbers

15. How to use the IMCSC function

The IMCSC function calculates the cosecant of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

What is a complex number?

A complex number contains a real and imaginary value, they let you for example solve equations with no real solutions like x² + 1 = 0  The chart above shows x² + 1 = 0 and it never touches the x-axis.

However, mathematicians invented the imaginary number and named it i, it extends into the complex plane and lets you solve equations using imaginary numbers.

What is the cosecant?

The cosecant is one of the six trigonometric functions, it is the multiplicative inverse of the sine function. This means that it is equal to 1 divided by the sine of a given angle θ.

The cosecant is csc θ = 1 / sin θ. In a right-angled triangle, the cosecant of an angle θ is equal to the hypotenuse divided by the opposite side.

What is the complex cosecant?

The complex cosecant is the extension of the cosecant function to the complex plane. It is defined as the reciprocal or the multiplicative inverse of the complex sine function, which means that it is equal to 1 divided by the sine of a complex number.

csc z = 1 / sin z, where z is a complex number.

The complex sine function can be expressed using the ordinary sine and cosine functions and the hyperbolic functions.

sin (x+yi) = sin x cosh y + i cos x sinh y

1. IMCSC Function Syntax

IMCSC(inumber)

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2. IMCSC Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMCSC function example

The image above demonstrates a formula in cell B28 that calculates the cosecant of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell D3:

The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.

Complex number 2+2i is the light blue line in the first quadrant. The cosecant of 2+2i is the small green line also displayed in the first quadrant.

3.1 Explaining formula

Step 1 - Populate arguments

IMCSC(inumber)

becomes

IMCSC(B25)

Step 2 - Evaluate the IMCSC function

IMCSC(B25)

becomes

IMCSC("2+2i")

and returns

0.244687073586957+0.107954592221385i

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4. How is the IMCSC function calculated in detail?

The cosecant of a complex number is calculated like this:

C = x + yi

csc(C) = (sin(x)*cosh(y) - icos(x)*sinh(y)) / (sin²(x)*cosh²(y) + icos²(x)*sinh²(y))

For example, C = 2 + 2i

csc(2+2i) = (sin(2)*cosh(2) - icos(2)*sinh(2)) / (sin²(2)*cosh²(2) + icos²(2)*sinh²(2))

becomes

csc(2+2i) = (3.42095486111701 - (-1.50930648532362i)) / 13.9809382284401

equals

csc(2+2i) = 0.244687073586957+0.107954592221385i

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5. IMCSC function not working

The IMCSC function returns a #VALUE! error if the argument is a boolean value.

The IMCSC function returns a #NUM! error if the argument is an invalid complex number. The i is missing in cell B25 shown in the image above.

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Useful links

IMCSC function - Microsoft
Cosecant of Complex Number
Trigonometric functions - Wikipedia

16. How to use the IMCSCH function

What is the IMCSCH function?

The IMCSCH function calculates the hyperbolic cosecant of a complex number in x + yi or x + yj text format.

The letter j is used in electrical engineering to distinguish between the imaginary value and the electric current.

What is a complex number?

A complex number contains a real and imaginary value, they let you for example solve equations with no real solutions like x² + 1 = 0  The chart above shows x² + 1 = 0 and it never touches the x-axis.

However, mathematicians discovered the imaginary number i that solves equations into the complex plane.

What is the hyperbolic cosecant?

Hyperbolic functions are similar to ordinary trigonometric functions, but they use a different shape to define them.

Trigonometric functions use a circle, while hyperbolic functions use a hyperbola. The chart above shows a hyperbola and two asymptotes (dashed lines) where the intersection is at the center of the hyperbola. The chart below shows a circle containing the trigonometric functions.

1. IMCSCH Function Syntax

IMCSCH(inumber)

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2. IMCSCH Function Arguments

inumber

Required. A complex number in x+yi or x+yj text format.

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3. IMCSCH function example

The image above demonstrates a formula in cell B28 that calculates the hyperbolic cosecant of a complex number specified in cell B25.

Cell C28 calculates the real number from the complex number in cell B28. Cell D28 extracts the imaginary number from the complex number specified in cell B28.

The real and imaginary numbers separated in a cell each allow us to graph the complex number on the complex plane.

Formula in cell D3:

The chart above shows the complex plane, the y-axis is the imaginary axis and the x-axis is the real axis.

Complex number 1+2i is the light blue line in the first quadrant. The hyperbolic cosecant of 1+2i is the green line displayed in the third quadrant.

3.1 Explaining formula

Step 1 - Populate arguments

IMCSCH(inumber)

becomes

IMCSCH(B25)

Step 2 - Evaluate the IMCSCH function

IMCSCH(B25)

becomes

IMCSCH("1+2i")

and returns

-0.221500930850509-0.6354937992539i

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4. How is the IMCSCH function calculated in detail?

The hyperbolic cosecant of a complex number is calculated like this:

csch(x + yi) = sinh(x)*cos(y) - icosh(x)*sin(y) / (sinh²(x)*cos²(y)+cosh²(x)*sin²(y))

For example, C=1+2i

csch(1 + 2i) = sinh(1)*cos(2) - icosh(1)*sin(2) / (sinh²(1)*cos²(2)+cosh²(1)*sin²(2))

becomes

csch(1 + 2i) = ( 1.1752011936438*-0.416146836547142 - i1.54308063481524*0.909297426825682 ) / (1.38109784554182*0.173178189568194+2.38109784554182*0.826821810431806)

becomes

csch(1 + 2i) = ( -0.489056259041293 - 1.40311925062204i ) / 2.20791965597362

equals

csch(1 + 2i) = -0.221500930850509-0.6354937992539i

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5. IMCSCH function not working

The IMCSCH function returns a #NUM! error if the provided argument is not a valid complex number.

The IMCSCH function returns a #VALUE! error if the provided argument is a boolean value.

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Useful links

IMCSCH function - Microsoft
Hyperbolic Cosecant of Complex Number
Hyperbolic functions

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