06 Feb Imaginary Numbers
Today we are going to talk about imaginary and complex numbers. Those of you who have taken some math tuition on advanced algebra will probably be familiar with the topic already. Imaginary numbers are one of the weirdest things in mathematics. Today, there are many theories which use imaginary numbers extensively. However, the situation was quite different when imaginary numbers were first introduced and many mathematicians were reluctant to use it. Further, many mathematicians thought that imaginary numbers are useless and they will not contribute in any way to the field of mathematics. The scenario however, after almost 500 years since imaginary numbers were first introduced, is quite different. From electronics to aviation, from number theory to quantum mechanics, imaginary numbers are used extensively and have become a make or break factor. Without further ado, let us see what they are and why they are so important.
What are imaginary numbers?
Imaginary numbers were first proposed by Gerolamo Cardano, an Italian mathematician of the 16th century. He was trying to figure out two numbers which, when added, yielded 10 and when multiplied, yielded 40. In other words, he was trying to find out a solution for x in the following equation:
x(10-x) -40 =0
He came up with the values of x as (5+√-15) and (5 – √-15). Indeed, if we add these numbers, we get 10 :
5 +√-15 + 5 -√-15 = 10
When we multiply, we get 40.
(5+√-15) (5 – √-15) = (5)2 – (√-15)2 = 25- (-15) = 25+15=40
So, we get the desired result. The problem arises when we try to find the value of the part (√-15) in the numbers. How can we find the square root of a number that is negative!!! Many mathematicians had been stumped over by this problem. Cardano, however, kept this part separate and did not try to find a value for the number -√-15. Instead, he just used it to find a solution to his problem. It is important to mention here that although he came up with the idea, Cardano himself did not like the idea of imaginary numbers very much.
Nowadays, we use a slightly different notation for imaginary numbers. The number (√-1) is known as the imaginary number and it is expressed by the symbol ‘i’. So, rather than writing (√-15), we would write the number as (√15i) where “i” signifies the imaginary part which is (√-1).
Complex numbers are those numbers which have an imaginary part associated with it. For example, the numbers (5+√15i) and (5-√15i) are known as complex numbers. Each complex number has two parts: the real part and the imaginary part. For the examples above, the number 5 is known as the real part and the part 15i is known as the imaginary part. In general, any complex numbers take the following form:
a + i.b
where a is the real part and the ‘i.b’ is the imaginary part.
The introduction of complex numbers was a pioneering discovery in the mathematical world. Although it was not appreciated in the early days post its inception, the concept gradually caught on with the community of mathematicians. By introducing complex numbers, mathematicians were able to tackle a vast number of problems which could not have been solved otherwise.
Do you remember when was the first time you heard about imaginary numbers in your maths tuition class? Imaginary and complex numbers is taught more extensively in A-level maths tuition classes than in O-level maths tuition classes. Though this topic is being introduced in the secondary level maths tuition classes in Singapore, it is just an introduction to the topic.
When you will learn about complex numbers in your math tuition classes, you will have some firsthand experience with imaginary and complex numbers and hopefully this article will help you to be better prepared for them.