Squares and Square Roots

Squares & Square Roots

Introduction

you might have seen numbers a with a small 2 on the top like this  ” ² ” and another symbol like this √, so what are they these are called squares and square roots.
In this post we will learn what are square roots, how we use them and  what is the purpose of square roots.

What are squares and Square Roots

A square  is when we multiply a number by Itself
ex.

5*5=25, so we call 25 as the square of 5

Perfect squares

Perfect squares are numbers which are obtained by squaring a whole number. and the square has no decimals in it. ex

Square Roots

Square is like and inverse/reverse operation to the squares. What we do is we have to find the square root/the multiple of a square, it is also called the base.
ex.

 

What is the purpose of squares and square roots

1. When you go to find a new apartment, you will want to know the size of it. Newspaper and online advertisements usually give the dimensions of the apartment by only listing its area, such as 625 square feet. This can become hard to visualize. However, if you use the concept of square roots and convert this number in the form of 25*25 square feet, it gives you a better idea of how big the apartment is.
2. Square roots are also used to find the period of a pendulum.
3. Squares and square roots are used in the Pythagoras theorem(post coming soon).

Square roots are also used in building , technology, engineering etc.

Finding square roots of perfect squares

You can tell what the square root of 16 easily because it is a small and easy to calculate. (√16=4, 4*4=16) but what if I say you need to find the square root of 1024? This is a really big number but you need to find it we don’t even know the times table of 1024!, but relax, there are ways to find this out…

• Long division method
• prime factorization method
• Vedic math method

Long division method

1. First we put two big lines like this, somewhat like the division symbol…
   
2. then we put the square (number) inside the two lines like this
   
3. right above the number we start adding bars after every two numbers from right → left.(If a number is left out when you are drawing bars it does not matter just put a bar over the one number.)
4. then we check the nearest perfect square to the first bar from the left or if it is already a prefect square write its base/root number (fir ex. if nine was the first number in the left most bar we would write three as the divisor and the first number in the quotient).  then we minus the nearest square with the dividend (10 in this case) & we bring down the second pair of numbers.
   
5. now we double the quotient  and write it as our new divisor wit a dash on it’s right (→).
   
6. Now we put a number in the dash (ex. 63) and multiply it with the same number you put in the dash (ex. 60×0 or 61×1 etc.).
   so now let us try with 62×2 and see what get and the same with 64×4 so when we get the product of these number we have to check which is the nearest or equal to the dividend(124 in this case.)
   
7. as you see 62×2 as perfectly divided with 124 so our quotient is the square root and if you have like more than 4 numbers repeat this process from “step 5”.

Prime factorization method

prime factorization is the method of finding factors of numbers using prime numbers (which are
2, 3, 5, 7, 11) so now we are going to use this method to find a square root of a prefect square

 

prime factors: numbers that are divisible only by itself.

Factors: of a number are the numbers that are multiplied to get the original number

 

 

Finding square root using prime factorization

 

1.   First we put the square in bars like this…
2. now we have to check what prime number is divisible with the number(1024) and divide it with the prime numbers from left to right repeat this step until you get on in the bars…
3. now  we have to pair the numbers in the side and count each pair as one number and multiply it.

Vedic math method

Note: this method only works for some 3 digits numbers and some 4 digit numbers.

 

1. First we  write down the number separately

   1024

2. then we check the units place for the nearest square for the number then we write it down  under the units place and minus it with 10 and write the difference under the number…
   
   
3.  now we check the last next 2 numbers and write the nearest square to the number and repeat the number again under the other number
   
   (why I wrote the number 3 is because 9 is the nearest perfect square number to  10 so we will need to write the square root for the number and the same rule applies to the other side)
4. now we need to choose one of each number here we have 32 and 38 to find out which is the square root in these two numbers we need to take number ending with five in that series of numbers like in our case 35 is a number ending with five in our series 30-40 now we need to find the square of 35 to do it in a easy way just take the first number from the left and add it by one and multiply the two numbers and add 2 zeros next to it and add it with 25
   
   
   35²=(3×4)hundreds+25=1225
5. so 35 square is bigger than 1024 so 38 square will be obviously be bigger than 1024 so it must be 32 which is the square of 1024!
   so we have found the answer for √1024