Identities of Complex Numbers
A complex number is a number that is made up of both real and imaginary numbers. Here we talk about some properties of complex numbers. To familiarize more with them read this post.
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 i2=−1.
As a result, a complex number is a straightforward representation of the addition of two integers, 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.
- The multiplying of two conjugate complex numbers will produce a real number as well as a complex number.
- If x and y are real numbers and x+yi=0, then x=0 and y=0 are the same value.
- 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: z1+z2=z2+z1 , z1.z2=z2.z1
- The complex number follows the “associative law” of addition and multiplication, which is a mathematical rule: (z1+z2)+z3=z1+(z2+z3) ,(z1.z2).z3=z1(z2.z3)
- The “distributive law” applies to complex numbers: z1(z2+z3)=z1.z2+z1z3
- In other words, if the sum of two complex numbers is real, and the product of two complex numbers is also genuine, then these complex numbers are conjugated with one another.
- For any two complex numbers, say z1 and z2, then |z1+z2|≤|z1|+|z2|
- When two complex numbers are multiplied by their conjugate value, the output should be a complex number with a positive value.
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.
- (z1+z2)2=(z1)2+(z2)2+2z1×z2
- (z1−z2)2=(z1)2+(z2)2−2z1×z2
- (z1)2−(z2)2=(z1+z2)(z1−z2)
- (z1+z2)3=(z1)3+3(z1)2z2+3(z2)2z1+(z2)3
- (z1−z2)3=(z1)3−3(z1)2z2+3(z2)2z1−(z2)3
Identities of Complex Numbers – Example 1:
Find the sum of the complex numbers. z1=−3+i and z2=4−3i
z1 + z2 =(−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: (z1+z2)2=(z1)2+(z2)2+2z1×z2
(2+i)2 = (2)2+(i)2+(2×2×i)=4+i2+4i
Then: i2=−1 → 4+i2=4−1=3
Now: 4+i2+4i=3+4i
Identities of Complex Numbers – Example 3:
Solve the complex numbers (3−i)3.
First, use this formula: (z1−z2)3=(z1)3−3(z1)2z2+3(z2)2z1−(z2)3
(3−i)3 =(3)3−3(3)2(i)+3(i)2(3)−(i)3 =27−27i+9i2−i3
Then: i2=−1 → 9i2=−9
=27−27i−9−i3 =18−27i−i3
i3=−i → 18−27i−i3=18−27i−(−i)= 18−27i+i
Now: 18−27i+i=18−26i
Exercises for Identities of Complex Numbers
Simplify.
- (4+5i)2
- (12+5i)+(3+i2+6i)
- (20+7i)−(45i+12)
- (5−4i)2(3+3i)
- (i2−5i)3
- (6−i)2−(10+i)2
- −9+40i
- 14+11i
- 8−38i
- 147−93i
- 74+110i
- −64−32i
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