In an earlier post we introduced the concepts of: real numbers, complex numbers, vectors and matrices. Let’s spend a little more time on the real and complex numbers.
A real number tells us which way to go (left or right) and by how much, starting from zero, along the line of real numbers. A negative number means that we go left, while a positive number means that we go right. Thus, this equation
-5 +7
has a direct translation of:
On the real numbers line, starting from zero: go left 5 then go right 7
If you visualize this action, you will find yourself arriving at 2 units to the right of zero. This translates to +2. Hence,
-5 +7 = +2
The plus (+) sign is often ignored and the equation is often written as:
-5+7 = 2
The need for a complex number came about when trying to take the square root of a real, negative number. A complex number is a pair of two numbers which are often written as a combination of a real number and an imaginary number, but can also be referenced as a pair of two numbers. The real number tells us which way to go on the real axis and the imaginary number tells us which way to go on the imaginary axis. A negative real number means go to the left, and a positive real number means go to the right. For the imaginary number, a positive imaginary number means go up and a negative imaginary number means go down on the imaginary line.
A complex number can be written such as:
A+B*i — (Equation 1)
or as an ordered pair, a vector:
(A,B) — (Equation 2)
The “i” in (Equation 1) means imaginary. The “i” helps with identifying which part is imaginary. Thus
A+B*i is the same as B*i+A
In the notation of (Equation 2) the pair is an ordered pair, meaning that the order in which the complex number is marked, is important. Thus
(A,B) is NOT EQUAL to (B,A)
When noting complex numbers, form (Equation 1) is most often used. The “i” in (Equation 1) has an interesting property. Let’s work with a specific example now. Let’s look at real number 5 and the complex number 5*i. Number 5 means that starting from zero we go to the right 5 units, while 5*i means that starting from zero, we go up 5 units.
This is equivalent to saying that 5*i is obtained by going to the right five units and then rotating counter-clockwise 90-degrees. Thus, multiplication by “i”: rotates the position of a number by 90-degrees, in the counter-clocwise direction.
What is then:
+5*i*i = ?
Working our way from left to right we have:
1.From zero go right 5 units.
2.Rotate counter clockwise by 90-degrees (first i multiplication)
3.Rotate counter clockwise by 90-degrees (second i multiplication)
4.Arrive at 5 units to the left of zero.
Thus:
+5*i*i = -5
If instead of 5, we would use 1, the equation would translate into:
+1*i*i = -1, or i*i = -1
In conclusion, real numbers go to the left and right on the real axis. Complex numbers are pairs of real and imaginary numbers, with reals going left and right; and imaginary going up and down on the imaginary axis. The imaginary number is a real number multiplied by “i”. Multiplication by “i” is equivalent to taking the current number position and rotating it counter-clockwise by 90-degrees.
– Darian Muresan
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