The technique for multiplying two complex numbers is similar to that used when multiplying out two brackets.
For example, (x + 2)(x + 3) = x² + 2x + 3x + 6 = x²+ 5x + 6

Now consider the two complex numbers (3 + 2i) and (4 + 5i)
To multiply these together take the same approach.

This will give (3 × 4) + (3 × 5i) + (2i × 4) + (2i × 5i)
= 12 + 15i + 8i + 10i²
= 12 + 23i - 10 (Remember that 10i² = 10(-1) = -10)
= 2 + 23i

Thus (3 + 2i)(4 + 5i) = (2 + 23i)

Rule for multiplying complex numbers Use the technique for multiplying out two brackets. (a + ib)(c + id) = (ac - bd) + i(bc + ad)

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Example 1 - Multiplying two complex numbers

Multiply (1 + 2i) and (2 - 3i)

Answer:
(1 + 2i)(2 - 3i) = (1 × 2) + (1× (-3i)) + (2i × 2) + (2i × (-3i))
= 2 - 3i + 4i + 6 = 8 + i

Example 2

Multiply (4 - 2i) and (2 + 3i)

Answer:
(4 - 2i)(2 + 3i) = 8 +12i - 4i + 6 = 14 + 8i

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Exercise on multiplying two complex numbers

Square roots of a complex number

Earlier examples showed that the square roots of a negative real number could be found in terms of i in the set of complex numbers. It is also possible to find the square roots of any complex number.

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Example 3 - Finding the square roots of a complex number

Find the square roots of the complex number 5 + 12i

Answer:
Let the square root of 5 + 12i be the complex number (a + ib) so (a + ib)²= 5 + 12i
Note: the trick is to equate the real parts and the imaginary parts to give two equations, which can be solved simultaneously.
First of all multiply out the brackets.
(a + ib)(a + ib) = a² - b² + 2abi
So (a² - b²) + 2abi = 5 +12i
Equate the real parts to give a² - b² = 5 (call this equation 1).
Equate the imaginary parts to give 2ab = 12 (call this equation 2).
Rearranging equation 2 to make b the subject gives
Substitute in equation 1 to give 5 =
So a⁴ - 5a² - 36 = 0
Hence (a² - 9)(a² + 4) = 0 a² = -4 or 9
Since a then a² 0 and a² = -4 is impossible.
This leaves a² = 9 which gives a = 3
Substitute a = 3 in equation 2 to give b = 2
Substitute a = -3 in equation 2 to give b = -2
Hence the square roots are 3 + 2i and -3 - 2i

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Exercise on finding the square roots of a complex number

Multiplication by i
Multiplication by i has an interesting geometric interpretation.
The following examples should demonstrate what happens.

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Example 4

Take the complex number 1 + 7i and multiply it by i

Answer:
(1 + 7i)i = i + 7i² = -7 + i

Example 5

Take the complex number -6 - 2i and multiply it by i

Answer:
(-6 - 2i)i = -6i - 2i² = 2 - 6i

Example 6

Take the complex number 8 - 4i and multiply it by i

Answer:
(8 - 4i)i = 8i - 4i² = 4 + 8i

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In each of the examples it can be seen that the effect of multiplying a complex number by i is a rotation of the point on the Argand diagram through 90° or

Interactive multiplication

Interactive multiplication by i

Activity
Take the complex number 2 + 3i. Plot the results on an Argand diagram when this number is repeatedly multiplied by the complex number i.
Hence give a geometric interpretation of multiplication by i and check this interpretation with other complex numbers.