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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
Complex conjugate
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.

For example, the complex conjugate of is It is worth noting that the product of a complex number and its conjugate is a real number.
Substitute for and simplify
This fact is very useful when simplifying quotients whose denominators are complex numbers.
Concept

Irrational Conjugate

Let and be rational numbers, with irrational. The irrational conjugate of is obtained by switching the sign of the irrational term.
For example, the conjugate of is
Rule

Sum of Conjugates

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

This is true for both the complex conjugates of complex numbers in the form of and for irrational conjugates of numbers in the form of

Proof

Complex Conjugates
Let and 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.
Simplify

Definition of complex conjugate

Similar operations can be used for the right-hand side.
Simplify

Definition of complex conjugate

Since both sides simplify to the same expression, they are equal.


Proof

Irrational Conjugates
Let and be two irrational numbers, where and 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.
Simplify

Definition of irrational conjugate

Similar operations can be used for the right-hand side.
Simplify

Definition of irrational conjugate

Since both sides simplify to the same expression, they are equal.
Rule

Product of Conjugates

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

This is true both for the complex conjugates of complex numbers of the form and for irrational conjugates of numbers of the form

Proof

Complex Conjugates
Let and be two complex numbers. Consider the left and right hand side of the equality separately, starting with the left-hand side.
Simplify

Definition of complex conjugate

Similar operations can be used for the right hand-side.
Simplify

Definition of complex conjugate

Since both sides simplify to the same expression, they must be equal.


Proof

Irrational Conjugates
Let and be two irrational numbers, where and are all rational. Consider the left and right hand side of the equality separately, starting with the left-hand side.
Simplify

Definition of irrational conjugate

Similar operations can be used for the right-hand side.
Simplify

Definition of irrational conjugate

Since both sides simplify to the same expression, they must be equal.


Rule

Power of Conjugates

For a number and a positive integer the conjugate of is equal to the power of the conjugate of

This is true for both the complex conjugates of complex numbers in the form of and for irrational conjugates of numbers in the form of

Proof

This can be proved by mathematical induction, using the Product of Conjugates Property.
1
Show That the Statement Is True for
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The statement is true for since and

2
Assume That the Statement Is True for Some Natural Number
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Assume that for some positive integer

3
Show That the Statement Is True for the Next Number
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Multiply the assumption by and simplify both sides.

The last line is the statement for
4
Draw a Conclusion
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The statement is true for and if it is true for then it is also true for Therefore, by the principle of mathematical induction, the statement holds true for all natural numbers

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