3. Square root
Working out the square root of a number is the reverse operation of working out the square of a number. Geometrically speaking, finding the square root of a number is the same as finding the length of the side of a square whose area is equal to the number given.
The perfect square root or exact square root of a number is other number , such as if it is squared we obtain the number .
√a = b since b2 =a
The “√ ” symbol is called the “radical”symbol. (Technically, just the “check mark” part of the symbol is the radical; the line across the top is called the “vinculum”.) The expression ” √9 ” is read as “root nine”, “radical nine”, or “the square root of nine”.
Here are the square roots of all the perfect squares from 1 to 100.
This a video on perfect square roots and their geometric interpretation:
- The square root of a negative number doesn’t exist: e. g. doesn’t exist.
- The square root of zero is only zero.
- Every positive number has two square root with the same absolute value and opposite signs.
The whole square root of a is the greatest integer b whose square is less than a.
√a = b since b2 < a
We work out the remainder by subtracting b2 from a:
Remainder = a – b2
3.1. Finding the square root with decimals by approximation
Example: Find √6 to 4 decimal places
Since 22 = 4 and 32 = 9, we know that √6 is between 2 and 3. Let’s just make a guess of it being 2.5. Squaring that we get 2.52 = 6.25. That’s too high, so make the guess a little less. Let’s try 2.4 next. To find approximation to four decimal places we need to do this till we have five decimal places, and then round the result.
|Guess||Square of guess||High/low|
|2.45||6.0025||Too high but real close|
|2.4495||6.00005025||Too high, so between 2.449 and 2.4495|
|2.4494||5.99956036||Too low, so between 2.4494 and 2.4495|
|2.44945||5.9998053025||Too low, so between 2.44945 and 2.4495.|
This is enough since we now know it would be rounded to 2.4495 (and not to 2.4494).
3.1. The algorithm to find the square root of a natural number
To find the square root of a natural number -using a piece of paper and a pencil- you have to take the steps we will see in the following page. Each time you click on INIT you will get a new example.
3.2. The algorithm of the square root with decimal digits
Only the natural numbers that are square perfects have another natural number as a square root. In the rest of the cases, in order to find its square root more accurately and exactly, you have to get decimal digits. Click on the picture if you want to practice:
Recall you can check your solution. It is always verified that:
Radicand – Root2 = Remainder
But be careful with the remainder if you got decimal digits. The true remainder is the remainder you got in the algorithm divided by 1 followed by as many pairs of zeroes as decimal digits you got in the root.
3.3. Why this strange algorithm?
I’m sure that I will always be
A lonely number like root three
The three is all that’s good and right,
Why must my three keep out of sight
Beneath the vicious square root sign,
I wish instead I were a nine
For nine could thwart this evil trick,
with just some quick arithmetic
I know I’ll never see the sun, as 1.7321
Such is my reality, a sad irrationality
When hark! What is this I see,
Another square root of a three
As quietly co-waltzing by,
Together now we multiply
To form a number we prefer,
Rejoicing as an integer
We break free from our mortal bonds
With the wave of magic wands
Our square root signs become unglued
Your love for me has been renewed