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Complex Numbers : Further Mathematics (AS-Level) : Mathematics Content Knowledge


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Further Mathematics (AS-Level)
Complex Numbers
Question 1 of 8

1. How confident are you that

a. you understand the language of complex numbers?


Example

The basic idea with complex numbers is to use i to represent the square root of negative 1. This means that it is possible to square root all real numbers, positive or negative.

Using the surd rules allows us to simplify expressions like:

\sqrt{-25} =\sqrt{25}\times\sqrt{-1} =5i  

Note that convention dictates that the i is normally written after numbers but before algebraic terms. We usually write 5i and 2\sqrt{3i} but x + iy. However, when working with several algebraic complex numbers it is sometimes more practical to write the i after the algebraic part – so a+bi + c+di is clearer.

Complex numbers are formed by adding real numbers to imaginary numbers – to form a number in the form of x+iy.

The x is the real part of the complex number and the y is the imaginary part of the complex number.

What this might look like in the classroom

Simplify:

a. \sqrt{-125}        b. \sqrt{-12}        c. \sqrt{-49}


Solutions:

a. \sqrt{-125} =\sqrt{25}  \times \sqrt{5} \times  \sqrt{-1} = 5\sqrt{5i}

b. \sqrt{-12} = \sqrt{4} \times \sqrt{3} \times \sqrt{-1} = 2\sqrt{3i}

c. \sqrt{-49}  = \sqrt{49}  \times\sqrt{-1} = 7i

Taking this mathematics further

Thinking back over the sorts of mathematics one can carry out with “regular” numbers – and wondering how these operations could be carried out with complex numbers.

Making connections

The key here is the completeness that complex numbers allows the mathematician. It is often helpful to trace through a history of number in order to strengthen the need for imaginary numbers.

Counting starts with integers and all addition is possible, multiplication too is possible with just integers. But the two inverses – subtraction and division give a need for some different sorts of numbers. Subtraction draws us to negative numbers and division draws us to fractions.

In the same way there is something quite ugly about not being able to square root a negative number – and thus there is a sensible argument for creating a new form of number to enable these sorts of square roots.

Later study shows that complex numbers are complete – that the four operations, and powers and roots performed on complex numbers yield a complex answer and do not lead to any further “can’t” situations.

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