# 3 Ways to Find Square Root Without Calculator

Calculating a square root is easy if you're working with an integer. Otherwise, it is important to know that there is a logical process to be followed to systematically discover the square root of any number, even without using a calculator. However, you first need to understand the basic steps of multiplication, addition and division.

## Steps

### Method 1 of 3: Finding the Square Root of Integers

#### Step 1. Calculate the perfect square using multiplication

The square root corresponds to a value that, when multiplied by itself, gives the original number. Another way to define it is to think of it this way: "what number can I multiply by itself to get the value in question?"

• For example, the square root of 1 is equal to 1 because 1 multiplied by 1 results in 1 (1×1=1). However, the square root of 4 is equal to 2, because 2 times 2 results in 4 (2×2=4). Think of the square root concept by imagining a tree. The tree can grow from a seed. Therefore, it is larger, but still related to the seed, which started at the height of the roots. In the example above, 4 represents the tree and 2 the seed.
• Consequently, the square root of 9 is equal to 3 (3×3=9), of 16 is equal to 4 (4×4=16), of 25 is equal to 5 (5×5=25), of 36 is equal to 6 (6×6=36), 49 equals 7 (7×7=49), 64 equals 8 (8×8=64), 81 equals 9 (9×9= 81) and 100 is equal to 10 (10×10=100).

#### Step 2. Use a division to find the square root

To find the square root of an integer, you can also divide that value by a few numbers until you get an answer identical to the one used in division.

• For example: 16 divided by 4 equals 4. And 4 divided by 2 equals 2, and so on. So in these examples, 4 is the square root of 16 and 2 is the square root of 4.
• Perfect roots don't have fractions or decimals because they involve whole numbers.

#### Step 3. Use the correct symbols to describe the square root

Mathematicians use a special symbol called a radical to indicate a square root. It looks like a tick symbol with a top line going to the right.

• N will represent the number whose square root you want to find, and must be inside the symbol used.
• So if you want to find the square root of 9, you must write a formula that puts the "N" (9) inside the symbol (the "radical") and has an equals sign and the number 3. That means "a square root of 9 equals 3".

### Method 2 of 3: Calculating the Square Root of Other Numbers

#### Step 1. Try to guess the value by elimination

It's harder to find non-integer square roots, but it's still possible.

• Suppose you want to find the square root of 20. You know that 16 is a perfect integer with square root equal to 4 (4×4=16). And likewise, 25 has a square root of 5 (5×5=25), so the square root of 20 should be these values.
• You could assume that the square root of 20 is 4, 5. Now, just square 4, 5 to check the assumption. This means that you need to multiply the number by itself: 4, 5×4, 5. See if the answer is above or below 20. If the guess is far from the expected result, try another number (maybe 4, 6 or 4, 4) and refine the guess to 20.
• For example, 4, 5×4, 5=20, 25. Logically, you should try a smaller number, probably going with 4, 4×4, 4=19, 36. So the square root of 20 should be between 4, 5 and 4, 4. How about we go with 4, 445×4, 445? The answer will be 19, 758, which is much closer. If you keep using different numbers in this process, you will finally come up with 4, 475×4, 475=20, 03. We round up, we get the number 20.

#### Step 2. Use the averaging process

This method also starts with your attempt to find the nearest whole numbers between which the desired value will be.

• Then divide the number by one of the square roots. Take the answer, calculate the mean and the value by which the division was done (the mean is the sum of the two numbers divided by two). Then take the original number and divide it by the average obtained. Finally, average this answer with the first average obtained.
• Sound complicated? It might be easier to follow an example. The number 10 lies between the two perfect roots of 9 (3×3=9) and 16 (4×4=16). The square roots of these numbers are 3 and 4. So divide 10 by the first number, 3. You get the result 3, 33. Now average between 3 and 3, 33 by adding the two numbers together and dividing the sum by 2. You will get the result 3, 1623.
• Review the calculations by multiplying the answer (in this case 3, 1623) by itself. In fact, 3, 1623 multiplied by 3, 1623 will equal 10, 001.

### Method 3 of 3: Squared Negative Numbers

#### Step 1. Square negative numbers with the same process

Remember that a negative number squared results in a positive value. So we will get a positive number in this situation.

• For example, -5×-5=25. However, remember that 5×5=25. So the square root of 25 could be either -5 or 5. Basically, there are two square roots for this value.
• Likewise, 3×3=9 and -3×-3=9, so the square root of 9 is equal to 3 and -3. The positive number is known as the "root root" and is the only answer you need at this point.

#### Step 2. After all, use a calculator

It's good to understand how to do math calculations in your head, but there are plenty of online calculators available that specifically calculate the square root.

• You can also find the square root symbol on a conventional calculator.
• Virtual calculators just need you to enter the number whose square root you want to calculate and press a button. The computer itself will perform the calculation immediately.

## Tips

• It's a good idea to memorize some of the perfect first squares:

• 02 = 0, 12 = 1, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49, 82 = 64, 92 = 81, 102 = 100.
• Further on, learn these: 112 = 121, 122 = 144, 132 169, 142 = 196, 152 = 225, 162 = 256, 172 = 289, […].
• A little more fun: 102 = 100, 202 = 400, 302 = 900, 402 = 1600, 502 = 2500, […].