IB Maths AI 1.9 Definitions

This page contains our IB Maths AI definitions for 1.9. By learning each one of these definitions, you will fully cover the content for IB Maths AI 'Introduction to complex numbers'.

All Topic 1 Sub-topics 1.1: Scientific Notation 1.2: Arithmetic Sequences 1.3: Geometric Sequences 1.4: Interest 1.5: Introduction To Exponents & Logs 1.6: Handling Numbers 1.7: Use Of Technology (HL)1.8: Advanced exponents & logs (HL)1.9: Introduction to complex numbers (HL)1.10: Advanced complex numbers (HL)1.11: Introduction to matrices (HL)1.12: Advanced matrices

argument

The angle a complex number makes with the positive real axis on an Argand diagram; for

z=r\operatorname{cis}\theta

, it is

\arg\left(z\right)=\theta

Cartesian form

A way to write a complex number as

z=a+bi

with

a,b\in\mathbb{R}

, separating its real and imaginary components.

cis

Shorthand for

\cos\theta+i\sin\theta

, so a complex number in polar form can be written as

z=r\operatorname{cis}\theta

complex

A number that can be written as

a+bi

with

a,b\in\mathbb{R}

, extending the real numbers by including multiples of

i

conjugate

The complex number obtained by changing the sign of the imaginary part; if

z=a+bi

then

\overline{z}=a-bi

, which reflects the point in the real axis on an Argand diagram.

A theorem stating that for a positive integer

n

\left[r\left(\cos\theta+i\sin\theta\right)\right]^n=r^n\left(\cos n\theta+i\sin n\theta\right)

, enabling powers and roots to be found using modulus and argument.

Euler

A numerical method that generates successive approximations to a solution by stepping forward with a fixed step size

h

using update rules such as

x_{n+1}=x_n+h y_n

imaginary

A number involving

i

; in

a+bi

this corresponds to the coefficient

b

i

, and if

a=0

the number is purely imaginary.

plane

A two-dimensional coordinate system used to represent complex numbers, with the horizontal axis as the real axis and the vertical axis as the imaginary axis.

quotients

Results of dividing one complex number by another; in Cartesian form, division can be carried out by multiplying numerator and denominator by the conjugate of the denominator to remove

i

from the denominator.

real

A number on the real number line, with no imaginary component; in

a+bi

this corresponds to

b=0

roots

Values of

w

that satisfy an equation of the form

w^n=z

; for

z=r\operatorname{cis}\theta

, the

n

th roots are

w_k=\sqrt[n]{r}\operatorname{cis}\left(\frac{\theta+2k\pi}{n}\right)

for

k=0,1,\dots,n-1

theorem

A mathematical result that can be proved logically from definitions and earlier results, then used to solve problems.

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