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Reference

Conjugation

Concept

Conjugate

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+bsqrt(c) a-bsqrt(c)
a-bsqrt(c) a+bsqrt(c)
Complex conjugate a+bi a-bi
a-bi a+bi
Concept

Complex Conjugate

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.


a+bi = a-bi or a-bi = a+bi

For example, the complex conjugate of z = 3 - 5i is z= 3 + 5i. It is worth noting that the product of a complex number and its conjugate is a real number.
z* z
â–Ľ
Substitute a+bi for z and simplify
Substitute

z= a+bi

( a+bi)(a+bi)
Substitute

a+bi= a-bi

(a+bi)( a-bi)
Multiply

Multiply

a^2-abi+abi-b^2i^2
AddTerms

Add terms

a^2-b^2i^2
IDefPow

i^2=- 1

a^2-b^2(-1)
NegNeg

- (- a)=a

a^2+b^2
This fact is very useful when simplifying quotients whose denominators are complex numbers.
Concept

Irrational Conjugate

Let a, b, and c be rational numbers, with sqrt(c) irrational. The irrational conjugate of a+bsqrt(c) is obtained by switching the sign of the irrational term. ccc Number & & Conjugate [0.15cm] a+bsqrt(c) & switch sign & a-bsqrt(c) [0.15cm] a-bsqrt(c) & switch sign & a+bsqrt(c)

For example, the conjugate of 2-5sqrt(3) is 2+5sqrt(3).
Rule

Sum of Conjugates

For two numbers z_1 and z_2, the conjugate of the sum is equal to the sum of the conjugates.


z_1+z_2=z_1+z_2

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+bsqrt(r).

Proof

 Complex Conjugates
Let z_1=a+bi and z_2=c+di be complex numbers. Consider the left- and right-hand sides of the equality separately. Then, beginning with the left, substitute the complex numbers and simplify each side.
z_1+z_2
SubstituteII

z_1= a+bi, z_2= c+di

( a+bi)+( c+di)
â–Ľ
Simplify
RemovePar

Remove parentheses

a+bi+c+di
CommutativePropAdd

Commutative Property of Addition

a+c+bi+di
FactorOut

Factor out i

(a+c)+(b+d)i

Definition of complex conjugate

(a+c)-(b+d)i
Similar operations can be used for the right-hand side.
z_1+z_2
SubstituteII

z_1= a+bi, z_2= c+di

a+bi+c+di
â–Ľ
Simplify

Definition of complex conjugate

(a-bi)+(c-di)
RemovePar

Remove parentheses

a-bi+c-di
CommutativePropAdd

Commutative Property of Addition

a+c-bi-di
FactorOut

Factor out - i

(a+c)-(b+d)i
Since both sides simplify to the same expression, they are equal.


z_1+z_2=z_1+z_2


Proof

 Irrational Conjugates
Let z_1=a+bsqrt(r) and z_2=c+dsqrt(r) be two irrational numbers, where a, b, c, d, and r are all rational. Consider the left- and right-hand sides of the equality separately. Then, beginning with the left, substitute the complex numbers and simplify each side.
z_1+z_2
SubstituteII

z_1= a+bsqrt(r), z_2= c+dsqrt(r)

( a+bsqrt(r))+( c+dsqrt(r))
â–Ľ
Simplify
RemovePar

Remove parentheses

a+bsqrt(r)+c+dsqrt(r)
CommutativePropAdd

Commutative Property of Addition

a+c+bsqrt(r)+dsqrt(r)
FactorOut

Factor out sqrt(r)

(a+c)+(b+d)sqrt(r)

Definition of irrational conjugate

(a+c)-(b+d)sqrt(r)
Similar operations can be used for the right-hand side.
z_1+z_2
SubstituteII

z_1= a+bsqrt(r), z_2= c+dsqrt(r)

a+bsqrt(r)+c+dsqrt(r)
â–Ľ
Simplify

Definition of irrational conjugate

(a-bsqrt(r))+(c-dsqrt(r))
RemovePar

Remove parentheses

a-bsqrt(r)+c-dsqrt(r)
CommutativePropAdd

Commutative Property of Addition

a+c-bsqrt(r)-dsqrt(r)
FactorOut

Factor out - sqrt(r)

(a+c)-(b+d)sqrt(r)
Since both sides simplify to the same expression, they are equal.


z_1+z_2=z_1+z_2

Rule

Product of Conjugates

For two numbers z_1 and z_2, the conjugate of the product is equal to the product of the conjugates.


z_1* z_2=z_1* z_2

This is true both for the complex conjugates of complex numbers of the form a+bi and for irrational conjugates of numbers of the form a+bsqrt(r).

Proof

 Complex Conjugates
Let z_1=a+bi and z_2=c+di be two complex numbers. Consider the left and right hand side of the equality separately, starting with the left-hand side.
z_1* z_2
SubstituteII

z_1= a+bi, z_2= c+di

( a+bi)*( c+di)
â–Ľ
Simplify
MultPar

Multiply parentheses

ac+adi+bci+bdi^2
IDefPow

i^2=- 1

ac+adi+bci-bd
CommutativePropAdd

Commutative Property of Addition

ac-bd+adi+bci
FactorOut

Factor out i

(ac-bd)+(ad+bc)i

Definition of complex conjugate

(ac-bd)-(ad+bc)i
Similar operations can be used for the right hand-side.
z_1* z_2
SubstituteII

z_1= a+bi, z_2= c+di

(a+bi)* (c+di)
â–Ľ
Simplify

Definition of complex conjugate

(a-bi)* (c-di)
MultPar

Multiply parentheses

ac-adi-bci+bdi^2
IDefPow

i^2=- 1

ac-adi-bci-bd
CommutativePropAdd

Commutative Property of Addition

ac-bd-adi-bci
FactorOut

Factor out - i

(ac-bd)-(ad+bc)i
Since both sides simplify to the same expression, they must be equal.


z_1* z_2=z_1*z_2


Proof

 Irrational Conjugates
Let z_1=a+bsqrt(r) and z_2=c+dsqrt(r) be two irrational numbers, where a, b, c, d, and r, are all rational. Consider the left and right hand side of the equality separately, starting with the left-hand side.
z_1* z_2
SubstituteII

z_1= a+bsqrt(r), z_2= c+dsqrt(r)

( a+bsqrt(r))*( c+dsqrt(r))
â–Ľ
Simplify
MultPar

Multiply parentheses

ac+adsqrt(r)+bcsqrt(r)+bd(sqrt(r))^2
PowSqrt

( sqrt(a) )^2 = a

ac+adsqrt(r)+bcsqrt(r)+bdr
CommutativePropAdd

Commutative Property of Addition

ac+bdr+adsqrt(r)+bcsqrt(r)
FactorOut

Factor out sqrt(r)

(ac+bdr)+(ad+bc)sqrt(r)

Definition of irrational conjugate

(ac+bdr)-(ad+bc)sqrt(r)
Similar operations can be used for the right-hand side.
z_1* z_2
SubstituteII

z_1= a+bsqrt(r), z_2= c+dsqrt(r)

(a+bsqrt(r))* (c+dsqrt(r))
â–Ľ
Simplify

Definition of irrational conjugate

(a-bsqrt(r))* (c-dsqrt(r))
MultPar

Multiply parentheses

ac-adsqrt(r)-bcsqrt(r)+bd(sqrt(r))^2
PowSqrt

( sqrt(a) )^2 = a

ac-adsqrt(r)-bcsqrt(r)+bdr
CommutativePropAdd

Commutative Property of Addition

ac+bdr-adsqrt(r)-bcsqrt(r)
FactorOut

Factor out -sqrt(r)

(ac+bdr)-(ad+bc)sqrt(r)
Since both sides simplify to the same expression, they must be equal.


z_1* z_2=z_1*z_2


Rule

Power of Conjugates

For a number z and a positive integer n, the conjugate of z^n is equal to the nth power of the conjugate of z.


z^n=(z)^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+bsqrt(r).

Proof
This can be proved by mathematical induction, using the Product of Conjugates Property.
1
Show That the Statement Is True for n=1
expand_more

The statement is true for n=1, since z^1=z and (z)^1=z.

2
Assume That the Statement Is True for Some Natural Number k
expand_more

Assume that z^k=(z)^k for some positive integer k.

3
Show That the Statement Is True for the Next Number k+1
expand_more
Multiply the assumption by z and simplify both sides.
z^k=z^k
MultEqn

LHS * z=RHS* z

z^k*z=z^k*z

z_1 * z_2=z_1* z_2

z^k* z=z^k*z
MultBasePow

a*a^m=a^(1+m)

z^(k+1)=z^(k+1)
The last line is the statement for n=k+1.
4
Draw a Conclusion
expand_more

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.

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