More Squares and Square Roots
Let us look at some interesting facts about squares and square roots.
A square and its square root are the reverse of each other. But what happens to squares and square roots of negative numbers?
When we square a negative number we get a positive result. This is because a negative times a negative gives a positive number.
For example 3 x 3 = 9 or 9 = 32
-3 x -3 = 9 or 9 = -32
Now what happens to the square root of any number? If I say the square of 3 is 9, and the square of -3 is also 9, what can you say about the square root of 9? We conclude that the square root of 9 is either 3 or -3.
Which means we have to keep in mind that every perfect number will have two answers for the root, positive and negative.
And when we speak about the square root of a negative number it is supposed to be imaginary number. These imaginary numbers are dealt with under a separate topic called complex numbers.
Square root of decimal numbers:
The square root has half the number of decimal places the square has.
For example: 35 x 35 = 1225 or 1225 = 352
And 3.5 x 3.5 = 12.25 or 12.25 = 3.52
So if the square has two decimal places then the square root has only one.
Now can we try to guess the square root of numbers that are not perfect squares? For example, find the square root of 10.
We have been taught the usual long division method in schools. But today we are going to guess the answer. We know that 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.
Let's try 3.5: 3.5 × 3.5 = 12.25
Let's try 3.2: 3.2 × 3.2 = 10.24
Let's try 3.1: 3.1 × 3.1 = 9.61
From this we come to a new conclusion that it lies between 3.1 and 3.2. We can start trying,
3.15 x 3.15 = 9.9225
3.16 x 3.16 = 9.9856
3.17 x 3.17 =10.0489
This gives an answer up to two decimal points. It may be a lengthy procedure but guessing helps you to identify and do certain calculations in a short cut method. We are getting closer to 10, but it will take a long time to get a good answer! But when we go for a competitive exam and the choices are given we can try guessing like this. It will save our time.
The 4th of April 2016 is a Square Root Day, because the date is 4/4/16. The next after that is the 5th of May 2025 (5/5/25).
Tip of the week: Learn short cuts for speedy calculations.