$$\DeclareMathOperator{cosec}{cosec}$$

# Number and Algebra

Syllabus Content

Complex numbers: the number i, where i2=-1. Cartesian form z=a+bi; the terms real part, imaginary part, conjugate, modulus and argument. The complex plane.

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## Furthermore

Official Guidance, clarification and syllabus links:

The complex plane is also known as the Argand diagram.

Formula Booklet:
 Complex numbers $$z=a+bi$$

Geometric Sequence:

$$i, i^2, i^3, i^4, i^6, i^7 ...$$ $$=i, -1, -i, 1, i, -1 ...$$

### Key Properties: $$(z^*)^*=z$$ $$(z_1+z_2)^*=z_1^* + z_2^*$$ $$(z_1-z_2)^*=z_1^* - z_2^*$$ $$(z_1z_2)^*=z_1^* \times z_2^*$$ $$\left( \dfrac{z_1}{z_2} \right)^* = \dfrac{z_1^*}{z_2^*}, \quad z_2 \neq 0$$ $$(z^n)^* = (z^*)^n \text{ for positive integers } n$$ $$z + z^* \text{ and } zz^* \text{ are real}$$ $$|z^*|=|z|$$ $$|z|^2 = zz^*$$ $$|z_1z_2|=|z_1||z_2|$$ $$\left| \dfrac{z_1}{z_2} \right| = \dfrac{|z_1|}{|z_2|}$$

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