Real and Imaginary Numbers in Mathematics

 

In mathematics most people are aware there are two kinds of numbers, namely positive numbers (+ 3, + 28.5) and negative numbers (- 3, – 28.5). Negative numbers were introduced in the Middle Ages to take care of problems like 8 minus 10.

For instance, in ancient times it seemed impossible for people to subtract 15 coins from 8 coins. The medieval bankers, however, had a clear notion of debt. “Give me ten fishes.  I only have money for four, but I will owe you for six,’ which is like saying (+4) – (+10) = (- 6).

However, positive and negative numbers can be multiplied if certain procedures are followed. A positive number multiplied by a positive number gives a positive number. A positive number multiplied by a negative number gives a negative number. But, most importantly, a negative number multiplied by a negative number gives a positive number.

Thus, (+ 2) x (+ 3) = (+6) and (+4) x (-6) = (-24) and (-5) x (-4) = (+20)

But should  we  ask ourselves what number multiplied by itself gives us +1?  One is + 1, since (+1) x (+1) = (+ 1). The other answer is -1, since (-1) x (-1) = (+1). Mathematicians put this into a square root symbol as √ +1 = ± 1.

But if we were to be  asked  what then is the square root of -1? Now we are in trouble. The answer

isn’t +1, because that multiplied by itself is +1.  The answer isn’t -1 either, because that multiplied by

itself is also +1.  Then we start to manipulate the values in another way by multiplying (+1) x (-1) =

(-1).  But that is multiplying two different numbers and not a number multiplied by itself.

In that case people invent a number and give it a special sign, let’s  say * 1, defining it as follows: *1 is

a number such that (*1) x (*1) = (-1). When this concept was first introduced, mathematicians called.

it as an ‘Imaginary number’ simply because it didn’t exist in the system of numbers to which they were

familiar. Actually, it is not as imaginary as the ordinary real numbers.  The so-called imaginary numbers

so that it can be manipulated easily with other real numbers.

Mathematicians then gave the ‘imaginary numbers’ the symbol ‘i’ so that we can have

positive imaginary numbers as + I, and imaginary numbers as - i, in which +1 is a positive real

number and - 1.   a negative real number. Thus, we can express √ -1 = ± i.

The system of real numbers can be exactly equated in the system of imaginary numbers. For instance

if we can have + 8, 32.68, + 7 /15, we can also have 8 i, 32.68 i + 7i /15

We can even have the real numbers on one side, and the imaginary system of numbers on the other.

For instance, we can represent the real number system on a straight line with 0 (zero) in the centre.

The positive numbers are placed on the one side of the zero and the negative numbers are on the

other.

We can then characterise the imaginary system of numbers along another line, crossing the first at

right angles at zero point, with the positive imaginaries on one side of the zero and the negative

imaginaries on the other. Numbers can be located anywhere in the plane by using both kinds together,

such as +2 +3i, or +3 + -2 i. These are called ‘complex numbers’.

Mathematicians and physicists find applications to be able to associate all the points in a plane with

a number system. They would not be able to do it without the imaginary numbers.