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Demystifying Complex Conjugates

February 2, 2012 GB High School Mathematics

This is the third post in the Complex Numbers Primer. In the previous post in this series, we have learned that complex numbers can be added and multiplied just like binomials. We did not discuss subtraction since it follows from addition; that is, if $a + bi$ and $c + di$ are complex numbers, then

$a + bi - (c + di) = a - c + (b - d)i$ .

Since we have completed the three fundamental operations on complex numbers, the next logical question would be how do we divide complex numbers? For example, how do we find the quotient of the expression

$\displaystyle\frac{5 + 2i}{3 - i}$ ?

Before we answer that question, recall that in simplifying rational expressions with radicals in the denominator, we multiply the numerator and the denominator of the expression by the ‘conjugate’ of the denominator. For instance, to simplify

$\displaystyle\frac{5}{2 + \sqrt{3}}$ . » Read more

One comment complex conjugates, complex numbers, real numbers

What is i? What are imaginary numbers?

January 24, 2012 GB High School Mathematics

Squaring numbers always give a positive result or a $0$ . For example, $3^2 = 9$ , $0^2 = 0$ , $(-5)^2 = 25$ . Using this argument, it follows that we can always get the square root of $0$ or any positive number. Now, what about the square root of a negative number, say, $-4$ ?

Let’s see: $2 \times 2 = 4$ and $-2 \times -2 =4$ ; therefore, no number exists that when multiplied by itself equals $-4$ . In fact, it looks like we cannot find the square root of any negative number.

Now, what if — just what if — we invent square root of negative numbers? Nobody would stop us right? We start with $-1$ . Suppose $- 1$ has a square root, and we call it $i$ . If $\sqrt{-1} = i$ , then $i^2 = -1$ . If so, we can also answer the question that we have asked above: $\sqrt{-4} = \sqrt{4(-1)} = 2i$ . As a matter of fact, we can generalize this operation for any negative number. If $a$ is positive, then $-a$ is a negative number. So, $\sqrt{-a} = \sqrt{a(-1)} = i\sqrt{a}$ .

But what kind of number is $i$ ? Surely, it is not a real number since we cannot locate it on the real number line. And, since we have all the real numbers and $i$ is not a real number, maybe, we should call $i$ an imaginary number.

The Real Number Line

Imaginary numbers were ‘invented’ (or discovered if you prefer) because mathematicians wanted to know if they could think of square root of negative numbers, particularly, the root of the equation $x^2 + 1 = 0$ (that is, $x^2 = - 1$ which is the same as finding the $\sqrt{-1}$ ). Just like the many ‘radical’ ideas in mathematics, it was not widely accepted at first, but eventually its invention proved to be very useful and has opened a lot of new ideas in mathematics.

In the next post in this series, we are going to discuss about the operations on imaginary numbers. .

8 comments imaginary number, real numbers, root of x^1 + 1, square root, what are imaginary numbers, what is i

Subset: a set contained in a set

September 30, 2011 GB College Mathematics, Set Theory

Two weeks ago, we have talked about the basics of sets. In this post, we are going to talk about subsets.

If you understand what this means, then you have a notion of a subset.

In mathematics, if $A$ and $B$ are sets, we say that $A$ is a subset of $B$ , denoted by $A \subseteq B$ , if all elements of $A$ are also elements of $B$ . The easiest way to illustrate this is through a Venn diagram as shown on the right. In the Venn diagram, set $A$ is within set $B$ . Therefore, all elements of set $A$ are also elements of $B$ .

Example 1

Let $B$ be set of all letters in the English alphabet and $A$ be the set of vowel letters. It is clear that $A \subseteq B$ since {a, e, i, o, u} are elements $A$ and also are elements of $B$ . » Read more

5 comments quadrilaterals, real numbers, sets, subsets, universal sets

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