Complex Number Operations

Number Theory and Algebra

• Complex Number Operations
  • Complex add
  • Complex subtract
  • Complex multiply
  • Multiplication of n complex numbers
  • Complex divide
  • Complex reciprocal
  • Complex absolute value
  • Complex square
  • Complex square root
  • Complex natural logarithm (base e)
  • Complex exponential
  • Complex exponential (t^{a + 𝒾b})
  • Integral power of a complex number
  • Integral roots of a complex number
  • Complex number to a complex power
  • Complex root of a complex number
  • Logarithm of a complex number to a complex base

Source: HP-45 Applications Book (HP 00045-90320 Rev. B Reorder 00045-66001, Dec 1974)

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Complex Number Operations

Note:

In this section, x and z are complex numbers (with x = a+𝒾b and z = u+𝒾v) and r and θ are used for polar decomposition of a complex number.

Complex add

Formula:

(a₁ + 𝒾b₁) + (a₂ + 𝒾b₂) = (a₁ + a₂) + 𝒾(b₁ + b₂) = u+𝒾v

Example:

(3 + 4𝒾) + (7.4 - 5.6𝒾) = 10.40 - 1.60𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	x2	+	z

Complex subtract

Formula:

(a₁ + 𝒾b₁) - (a₂ + 𝒾b₂) = (a₁ - a₂) + 𝒾(b₁ - b₂) = u+𝒾v

Example:

(3 + 4𝒾) - (7.4 - 5.6𝒾) = -4.4 + 9.6𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	x2	-	z

Complex multiply

Formula:

(a₁ + 𝒾b₁)(a₂ + 𝒾b₂) = (a₁a₂ - b₁b₂) + (a₁b₂ + a₁b₂) = u+𝒾v

Example:

(3 + 4𝒾)(7 - 2𝒾) = 29.00 + 22.00𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	x2	×	z

Multiplication of n complex numbers

Formula:

where a_k + 𝒾b_k = r_k e^𝒾θ_k .

Example:

(3 + 4𝒾) (7 - 2𝒾) (4.38 + 7𝒾) (12.3 - 5.44𝒾) = 1296.66 + 3828.90 𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	xk	×	z	Perform 2 for k=2,3, …, n

Complex divide

Formula:

(a₁ + 𝒾b₁) ÷ (a₂ + 𝒾b₂) = (r₁/r₂) e^{𝒾(θ₁ - θ₂)} = u+𝒾v

where

a₁ + 𝒾b₁ = r₁ e^𝒾θ₁
a₂ + 𝒾b₂ = r₂ e^𝒾θ₂ ≠ 0

Example:

(3 + 4𝒾) ÷ (7 - 2𝒾) = 0.25 + 0.64𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	x2	÷	z

Complex reciprocal

Formula:

1 ÷ (a₁ + 𝒾b₁) = (a - 𝒾b) ÷ (a² + b²) = u+𝒾v

Example:

1 ÷ (2 + 3𝒾) = 0.15 - 0.23𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	1/x	z

Complex absolute value

Formula:

|a + 𝒾b| = √(a² + b²)

Example:

|3 + 4𝒾| = 5.00

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	ABS	r

Complex square

Formula:

(a + 𝒾b)² = (a² - b²) + 𝒾(2ab) = u+𝒾v

Example:

(7 - 2𝒾)² = 45.00 - 28.00𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	ENTER ×	z

Complex square root

Formula:

√(a + 𝒾b) = ± r^½ e^𝒾θ⁄₂

where

a+𝒾b = r e^𝒾θ

Example:

√(7 + 6𝒾) = ±(2.85 - 1.05𝒾)

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	ENTER 0 . 5 y˟	z

Complex natural logarithm (base e)

Formula:

ln(a + 𝒾b) = ln √(a² + b²) + 𝒾 atan(^b⁄_a) = ln r + 𝒾θ = u+𝒾v (θ is in radians)

Example:

ln 𝒾 = 1.57𝒾

Note: a + 𝒾b = r e^𝒾θ = r(cos θ+ 𝒾 sin θ)

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	LN	z

Complex exponential

Formula:

e^{a + 𝒾b} = e^a e^𝒾b = u+𝒾v

Example:

e^1.57𝒾 = 1.00𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	e˟	z

Complex exponential (t^{a + 𝒾b})

Formula:

t^{a + 𝒾b} = e^{(a + 𝒾b)ln t} = u+𝒾v (t > 0)

Example:

2^{3 + 4𝒾} = -7.46 + 2.89𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	t	ENTER
2	x	y˟	z

Integral power of a complex number

Formula:

(a + 𝒾b)ⁿ = rⁿ (cos nθ + 𝒾 sin nθ) = u+𝒾v

where r = √(a² + b²), θ = atan(^b⁄_a) and n is a positive integer.

Example:

(3 + 4𝒾)⁵ = -7.46 + 2.89𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	ENTER
2	n	y˟	z

Integral roots of a complex number

Formula:

(a + 𝒾b)^1⁄_n = r^1⁄_n (cos ^{(θ+360k)⁄_n} + 𝒾 sin ^{(θ+360k)⁄_n}) = u_k+𝒾v_k

where n is a positive integer and k = 0,1, …, n -1. (θ is in degrees)

Example:

5 + 3𝒾 has three cube roots:
u₀+𝒾v₀ = 1.77 + 0.32𝒾
u₁+𝒾v₁ = -1.16 + 1.37𝒾
u₂+𝒾v₂ = -0.61 - 1.69𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x	ENTER
2	n	1/x y˟	z0	k = 0
3		LASTx 2 ×
4		π × ×
5		e˟ ×	z1	k = 1
6		LASTx ×	zk	Perform 6 for k=2,3, …, n-1

Complex number to a complex power

Formula:

(a₁ + 𝒾b₁)^{(a₂ + 𝒾b₂)} = e^{(a₂ + 𝒾b₂)ln(a₁ + 𝒾b₁)} = u+𝒾v

where a₁ + 𝒾b₁ ≠ 0

Example:

(1 + 𝒾)^{(2 - 𝒾)} = 1.49 + 4.13𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	x2	y˟	z

Complex root of a complex number

Formula:

(a₁ + 𝒾b₁)^{1⁄_{(a2 + 𝒾b2)}} = e^{[ln(a₁ + 𝒾b₁) / (a₂ + 𝒾b₂)]} = u+𝒾v

where a₁ + 𝒾b₁ ≠ 0

Example:

Find the (2 - 𝒾)^th root of 1.49 + 4.13𝒾

Answer:

1.00 + 1.00𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	ENTER
2	x2	1/x y˟	z

Logarithm of a complex number to a complex base

Formula:

log_{(a1 + 𝒾b1)}(a₂ + 𝒾b₂) = ln(a₂ + 𝒾b₂) ÷ ln(a₁ + 𝒾b₁) = u+𝒾v

where a₁ + 𝒾b₁ ≠ 0

Example:

log_{(1 + 𝒾)}(1.49 + 4.13𝒾) = 2.00 - 1.00𝒾

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	x1	LN
2	x2	LN x<>y ÷	z