Conjugates
A pair of complex numbers with a different sign but otherwise the same values are called conjugates:
$$ 3 + 4i 3 - 4i $$
What are complex conjugates??
A pair of complex numbers with real and imaginary parts equal in magnitude but opposite in sign:
- $a+bi$ and $a-bi$
What is a pair of complex numbers $a+bi$ and $a-bi$ called??
A complex conjugate.
What is the result of multipling complex conjugates??
You get a real number.
What other topic links to multiplying complex conjugates??
The difference of two squares.
What is the result of adding complex conjugates??
- You get a real number.
- $(a+bi)+(a-bi) = 2a$
What is the result of subtracting complex conjugates??
- You get an imaginary number.
- $(a+bi)-(a-bi) = 2bi$
What is $z + z^{\ast} for $z=(a+bi)$??
$2a$.
What is $z - z^{\ast}$ for $z=(a+bi)$??
$2bi$.
What is $zz^{\ast}$ for $z=(a+bi)$??
$a^2 + b^2$.
Q: What is $(3 + 4i)(3 - 4i)$??
$$ 25 $$
Q: What is $(3 + 4i) + (3 - 4i)$??
$$ 6 $$
Q: What is $(3 + 4i) - (3 - 4i)$??
$$ 8i $$
What is the conjugate of a real number??
Itself.
What is $x^{\ast}$ where $x \in \mathbb{R}$??
$x$.
What does $z^{\ast}$ mean??
The complex conjugate of $z$.
What is the notation for the complex conjugate of $z$??
$z^{\ast}$
How can you divide complex numbers??
- Using complex conjugation to convert the denominator to a real number
- Multiply the top and bottom by the complex conjugate of the denominator
Backlinks
- [[Further Maths - Syllabus]]S
- [[Further Maths - Complex Numbers]]S
- [[Further Maths - Complex Numbers]]S
Metadata
date: 2020-09-07 19:07
tags:
- '@?further-maths'
- '@?school'
- '@?a-level'
- '@?public'
title: Further Maths - Conjugates