| {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} |
| {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} |
| {{ 'ml-lesson-time-estimation' | message }} |
The conjugate of an irrational binomial is the same number, but the sign of the irrational term is switched. Similarly, the conjugate of a complex number is the same complex number where the sign of the complex part is opposite to its original sign.
Binomial | Conjugate | |
---|---|---|
Irrational conjugate | a+bc | a−bc |
a−bc | a+bc | |
Complex conjugate | a+bi | a−bi |
a−bi | a+bi |
The complex conjugate of a complex number has the same real part, but the imaginary part is the opposite of its original sign. Therefore, changing the sign of the imaginary part of a complex number creates its complex conjugate. It is denoted by a line drawn above the complex number.
z=a+bi
a+bi=a−bi
Multiply
Add terms
i2=-1
-(-a)=a
For two numbers z1 and z2, the conjugate of the sum is equal to the sum of the conjugates.
z1+z2=z1+z2
z1=a+bi, z2=c+di
Remove parentheses
Commutative Property of Addition
Factor out i
Definition of complex conjugate
z1=a+bi, z2=c+di
Definition of complex conjugate
Remove parentheses
Commutative Property of Addition
Factor out -i
z1=a+br, z2=c+dr
Remove parentheses
Commutative Property of Addition
Factor out r
Definition of irrational conjugate
z1=a+br, z2=c+dr
Definition of irrational conjugate
Remove parentheses
Commutative Property of Addition
Factor out -r
For two numbers z1 and z2, the conjugate of the product is equal to the product of the conjugates.
z1⋅z2=z1⋅z2
z1=a+bi, z2=c+di
Multiply parentheses
i2=-1
Commutative Property of Addition
Factor out i
Definition of complex conjugate
z1=a+bi, z2=c+di
Definition of complex conjugate
Multiply parentheses
i2=-1
Commutative Property of Addition
Factor out -i
z1=a+br, z2=c+dr
Multiply parentheses
(a)2=a
Commutative Property of Addition
Factor out r
Definition of irrational conjugate
z1=a+br, z2=c+dr
Definition of irrational conjugate
Multiply parentheses
(a)2=a
Commutative Property of Addition
Factor out -r
For a number z and a positive integer n, the conjugate of zn is equal to the nth power of the conjugate of z.
zn=(zˉ)n
This is true for both the complex conjugates of complex numbers in the form of a+bi and for irrational conjugates of numbers in the form of a+br.
The statement is true for n=1, since z1=zˉ and (zˉ)1=zˉ.
Assume that zk=(zˉ)k for some positive integer k.
LHS⋅zˉ=RHS⋅zˉ
z1⋅z2=z1⋅z2
a⋅am=a1+m
The statement is true for n=1, and if it is true for n=k, then it is also true for n=k+1. Therefore, by the principle of mathematical induction, the statement holds true for all natural numbers n.