Complex Numbers and their Properties

Imaginary numbers as we have discussed in Tuesday’s post are numbers of the form bi where b is a real number and i = \sqrt{-1}. The term imaginary as (opposed to real) was first used by Rene Descartes, the mathematician who invented Coordinate Geometry — the Cartesian plane in particular.  Leonhard Euler was the one who introduced the symbol i for \sqrt{-1}.

leonard euler

Leonhard Euler

Gerolamo Cardano, a pioneer in probability, was the one who suggested the   use of numbers of the form a + bi where a and b are real numbers and i = \sqrt{-1}.  Numbers of this form were named complex numbers by Carl Frederich Gauss.The real part of a + bi is a and the imaginary part is bi.

Note that if we treat these numbers as binomials, then, to add 2 - 3i and 4 + 5i will result to (2+4) + (-3i+5i) = 6 + 2i .  Similarly, multiplying (6 - 5i)(3 + i) gives us

18- 9i - 5i^2 = 15 - 9i - 5(-1) = 23 - 9i.

The following follow from the operations that we have done above: (1) (a + bi) + (c+di) = (a +c) + (b+d)i and (2) (a + bi)(c+di) = (ac - bd)(ad + bc)i Prove this!

We can also observe that (a+bi)=(c+di) if and only if a = c and b = d.
In the next post, we are going to learn more about the properties of complex numbers.

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