Further Maths - Conjugates
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 $(3 + 4i)(3 - 4i)$?
Q: What is $(3 + 4i) + (3 - 4i)$?
Q: What is $(3 + 4i) - (3 - 4i)$?
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