Identities of Complex Numbers

A complex number is written as \(a+ib\) and usually represented by \(z\). Where \(a\) signifies a real number and \(ib\) represents an imaginary number. In addition, \(a,b\) are real values, and \(i^2 = -1\).

As a result, a complex number is the sum of a real part and an imaginary part, namely a real and an imaginary number. One side is entirely genuine, while the other is entirely imagined.

Related Topics

• How to Multiply and Divide Complex Numbers
• How to Add and Subtract Complex Numbers
• How to Solve Rationalizing Imaginary Denominators

A step-by-step guide to identities of complex numbers

The following are some of the properties of complex numbers:

• The sum of two conjugate complex numbers will result in a real number.
• Multiplying a complex number by its conjugate always gives a real number that is zero or positive.
• If (x) and (y) are real numbers and \(x+yi =0\), then (x =0) and (y =0).
• When two conjugate complex numbers are added together, the result is a real number.
• A real number can be obtained by multiplying two conjugate complex numbers.
• If \(p\), \(q\), \(r\), and \(s\) are real numbers, then \(p+qi = r+si\), \(p = r\), and \(q=s\).
• The “basic law” of addition and multiplication applies to complex numbers: \(\color{blue}{z_{1 }+ z_{2 }= z_{2 }+ z_{1 } }\), \(\color{blue}{z_{1 }. z_{2 }= z_{2 }. z_{1 } }\)
• The complex number follows the “associative law” of addition and multiplication, which is a mathematical rule: \(\color{blue}{(z_{1 }+ z_{2 })+ z_{3 }= z_{1 }+ (z_{2 }+z_{3 })}\),\(\color{blue}{(z_{1 }. z_{2 }).z_{3 }=z_{1 } (z_{2 }. z_{3 } )}\)

• The “distributive law” applies to complex numbers: \(\color{blue}{z_{1 }( z_{2 }+z_{3 })= z_{1 }. z_{2 }+z_{1 }z_{3 } }\)
• In other words, if the sum of two complex numbers is real, and the product of two complex numbers is also genuine, then these numbers are conjugates of each other, as long as they are not real numbers.
• For any two complex numbers, say \(z_{1 }\) and \(z_{2 }\), then \(|z_{1 }+z_{2 }|≤|z_{1 }|+|z_{2 }|\)
• When a complex number is multiplied by its conjugate, the output is a real number that is zero or positive.

Algebraic identities of complex numbers

All algebraic identities apply equally to complex numbers. The addition and subtraction of complex numbers with the exponents of \(2\) or \(3\) can be easily solved using algebraic identities of complex numbers.

• \(\color{blue}{(z_{1 }+ z_{2 })^2= (z_{1 })^2+ (z_{2 })^2+2 z_{1 }× z_{2 } }\)

• \(\color{blue}{(z_{1 }- z_{2 })^2= (z_{1 })^2+ (z_{2 })^2-2 z_{1 }× z_{2 } }\)

• \(\color{blue}{(z_{1 })^2- (z_{2 })^2= (z_{1 }+ z_{2 })( z_{1 }- z_{2 }) }\)

• \(\color{blue}{(z_{1 }+ z_{2 })^3= (z_{1 })^3+ 3(z_{1 })^2 z_{2 } +3 (z_{2 })^2 z_{1 }+ (z_{2 })^3 }\)

• \(\color{blue}{(z_{1 }- z_{2 })^3= (z_{1 })^3- 3(z_{1 })^2 z_{2 } +3 (z_{2 })^2 z_{1 }- (z_{2 })^3 }\)

Identities of Complex Numbers, Example 1:

Find the sum of the complex numbers. \(z_{1 }=-3+i\) and \(z_{2 }=4-3i\)

\(z_{1 }\) \(+\) \(z_{2 }\) \(=(-3+i)+(4-3i)=(-3+4)+(i-3i)=1-2i\)

Identities of Complex Numbers, Example 2:

Solve the complex numbers \((2+i)^2\).

To solve complex numbers use this formula: \(\color{blue}{(z_{1 }+ z_{2 })^2= (z_{1 })^2+ (z_{2 })^2+2 z_{1 }× z_{2 } }\)

\((2+i)^2\) \(=\) \((2)^2\)\(+\)\((i)^2\)\(+(2×2×i)\)\(=4+i^2+4i\)

Then: \(i^2=-1\) → \(4+i^2=4-1=3\)

Now: \( 4+i^2+4i =3+4i\)

Identities of Complex Numbers, Example 3:

Solve the complex numbers \((3-i)^3\).

First, use this formula: \(\color{blue}{(z_{1 }- z_{2 })^3= (z_{1 })^3- 3(z_{1 })^2 z_{2 } +3 (z_{2 })^2 z_{1 }- (z_{2 })^3 }\)

\((3-i)^3\) \(=(3)^3-3(3)^2(i)+3(i)^2(3)-(i)^3\) \(=27-27i+9i^2-i^3\)

Then: \(i^2=-1\) → \(9i^2=-9\)

\(=27-27i-9-i^3\) \(=18-27i-i^3\)

\(i^3=-i\) → \(18-27i-i^3= 18-27i-(-i)=\) \(18-27i+i\)

Now: \( 18-27i+i =18-26i\)

Exercises for Identities of Complex Numbers

Simplify.

1. \(\color{blue}{(4+5i)^2}\)
2. \(\color{blue}{(12+5i)+(3+i^2+6i)}\)
3. \(\color{blue}{(20+7i)-(45i+12)}\)
4. \(\color{blue}{(5-4i)^2(3+3i)}\)
5. \(\color{blue}{(i^2-5i)^3}\)
6. \(\color{blue}{(6-i)^2-(10+i)^2}\)

1. \(\color{blue}{-9+40i}\)
2. \(\color{blue}{14+11i}\)
3. \(\color{blue}{8-38i}\)
4. \(\color{blue}{147-93i}\)
5. \(\color{blue}{74+110i}\)
6. \(\color{blue}{-64-32i}\)

Conjugate and Modulus Identities

For z = a + bi, the conjugate is z̄ = a − bi and the modulus is |z| = √(a² + b²). These identities hold for all complex numbers z, z₁ and z₂ (with z₂ ≠ 0 when dividing).

Example: Let z = 3 + 4i. Then z̄ = 3 − 4i, and z · z̄ = 9 − 12i + 12i − 16i² = 9 + 16 = 25. So |z| = √25 = 5. This identity is also why we multiply by the conjugate to divide complex numbers: it turns the denominator into a real number.

Powers of i and Euler’s Formula

The powers of i repeat every four steps: i¹ = i, i² = −1, i³ = −i, i⁴ = 1. To find a large power, divide the exponent by 4 and use the remainder. For example, 23 ÷ 4 leaves a remainder of 3, so i²³ = i³ = −i.

Euler’s formula links complex numbers to trigonometry: e^(iθ) = cos θ + i sin θ. When θ = π, cos π = −1 and sin π = 0, so e^(iπ) = −1. Rearranged, this is Euler’s identity, e^(iπ) + 1 = 0.

A related rule is De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). It makes powers of complex numbers in polar form much faster than multiplying by hand.

Frequently Asked Questions

What are the identities of complex numbers?

They are equations that hold for every complex number z = a + bi. The usual algebraic identities still work, such as (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂². Others are special to complex numbers: i² = −1, z · z̄ = a² + b² = |z|², and |z₁z₂| = |z₁| · |z₂|. For example, (1 + 2i)² = 1 + 4i + 4i² = −3 + 4i.

What is the additive identity of complex numbers?

The additive identity is 0, written as 0 + 0i. Adding it to any complex number leaves that number unchanged: (4 − 7i) + (0 + 0i) = 4 − 7i. Every complex number a + bi also has an additive inverse, −a − bi, and adding the two gives the additive identity 0.

What is the multiplicative identity of complex numbers?

The multiplicative identity is 1, written as 1 + 0i. Multiplying any complex number by it gives the same number back: (3 + 5i)(1 + 0i) = 3 + 5i. Every nonzero complex number z also has a multiplicative inverse 1/z = z̄ / |z|², so for 1 + i the inverse is (1 − i)/2.

Can complex numbers be real numbers?

Yes. Every real number is a complex number whose imaginary part is zero. For instance, 8 can be written as 8 + 0i, so 8 is a complex number as well as a real number. The real numbers sit on the horizontal axis of the complex plane, and identities like z · z̄ = |z|² still hold for them.

Are all imaginary numbers complex numbers?

Yes. A pure imaginary number such as 6i can be written as 0 + 6i, which is a complex number with real part 0. The reverse is not true: a complex number like 2 + 3i is not purely imaginary because its real part is not zero. Pure imaginary numbers lie on the vertical axis of the complex plane.

What are the trigonometric identities of complex numbers?

The key one is Euler’s formula, e^(iθ) = cos θ + i sin θ, which lets you write z in polar form as r(cos θ + i sin θ). From it comes De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. Setting θ = π gives Euler’s identity, e^(iπ) + 1 = 0.