3BComplexArithmetic

Arithmetic with Complex Numbers

Arithmetic can be represented on the Complex Plane.

Complex Arithmetic on the Calculator

The calculator can do complex arithmetic:
 * Set your calculator to CPLX mode instead of REAL
 * Use i from the list of variables in the MTH tab of the virtual keyboard, not from the ABC keyboard
 * Be careful to use brackets when appropriate

Addition and Subtraction

Add/subtract the real parts together. Add/subtract the imaginary parts together.

Eg 1 ... ... (3 + 5i) + (4 – 2i) = 7 + 3i

Eg 2 ... ... (3 + 5i) – (4 – 2i) = –1 + 7i

Multiplication

To multiply by a constant (or scalar), expand brackets as per Year 9 algebra:

Eg ... ... 3(5 + 2i) = 15 + 6i

To multiply two complex numbers together, expand brackets as per Year 9 algebra:

Eg ... ... (3 + 5i)(4 – 2i) ... ... = 3×4 + 3×–2i + 5i×4 + 5i×–2i ... ... = 12 – 6i + 20i – 10i 2 ... ... {10i 2 becomes –10} ... ... = 22 + 14i

Multiplication by **i**

If ... ... z = x + yi Then ... ... zi = i(x + yi) ... ... ... = ix + yi 2 ... ... ... = –y + xi

On an Argand Diagram, this is the same as **z** but rotated 90° anti-clockwise around the centre.


 * Example **

Find zi and show on an Argand Diagram, the change from z to zi, when ... ... z = 3 + 2i


 * Solution:**

... ... zi = i(3 + 2i) ... ... ... = 3i + 2i 2 ... ... ... = –2 + 3i

... ... See diagram on right

Similarly, ... ... zi 2 = i 2 (x + yi) ... ... ... = –1(x + yi) ... ... ... = –x – yi

On an Argand Diagram, this is the same as **z** but rotated 180° anti-clockwise.

So zi n rotates **z** anti-clockwise by 90n degrees.

Division

To divide two complex numbers, multiply by the complex conjugate of the denominator over itself.

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