How to Find the Square Root Without Using a Calculator

You can easily find the square root of a number if the number is an integer. If the number is not an integer, you can use the following logic to find the square root of any number, even if you don’t have a calculator. You will need to know how to apply simple multiplication, addition, and division to do this.

Table of Contents

Find the square root of an integer

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Find the square root by multiplying. The square root of a number is the number that when you multiply the number by itself, you get the first number you have. Think of it like this: “What number can you multiply by itself to get the number you already have?”

For example, the square root of 1 is 1 because 1 times 1 equals 1 (1X1=1). However, the square root of 4 is 2 because 2 times 2 equals 4 (2X2=4). Imagine the concept of a square root as a tree. The tree sprouts from an oak seed. So even though the tree is larger than the seed, the tree is still related to the seed because the seed is the root of the tree. In the above example, 4 is the tree and 2 is the acorn.
So the square root of 9 is 3, of 16 is 4 (4X4=16), of 25 is 5 (5X5=25), of 36 is 6 (6X6=36), of 49 is 7 (7X7=49). , of 64 is 8 (8X8=64), of 81 is 9 (9X9=81), and of 100 is 10 (10X10=100). ^{[1] X Research Source}

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Use division to find the square root. To find the square root of an integer, you can divide the integer by numbers in turn until you find a quotient that is exactly the same as your divisor.

For example: 16 divided by 4 equals 4. 4 divided by 2 equals 2, and so on. So in the above example, 4 is the square root of 16, 2 is the square root of 4.
Squares are not fractions or decimals because they are square roots of integers.

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Use the correct symbol for the square root. Mathematicians use a special symbol called a radical sign to write square roots. This mark will look like a tick connected to a horizontal line. ^{[2] X Research Source}

N is the number for which you are looking for the square root. N will be in the radical sign. ^{[3] X Research Sources}
So if you’re trying to find the square root of 9, your formula will have N(9) in the root sign, then the equal sign, and the number 3. This formula means “the square root of 9 equals 3. .”

Find the square root of other numbers

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Guess then use the elimination method. It will be harder for you to find the square root of a non-integer, however, that doesn’t mean you can’t.

For example, you want to find the square root of 20. You already know that 16 is a perfect square with the square root of 4 (4X4=16). And 25 also has a square root of 5 (5X5=25), so the square root of 20 should be in that range.
You can guess that the square root of 20 is 4.5. Now square 4.5 to test your prediction. That is, you multiply 4.5 by itself: 4.5X4.5. Notice if the answer is greater than or less than 20. Then keep multiplying by the prediction (4.6 if the result is greater than 20 or 4.4 if less than 20) and keep predicting until when you come up with a result of 20. ^{[4] X Research Source}
For example, 4.5X4,5 = 20.25, in theory you should try a smaller number, maybe 4.4. 4,4X4,4 = 19.36. So the square root of 20 would be between 4.5 and 4.4. Try 4,445X4,445. The answer is 19,758. That’s almost 20 already. If you keep trying different numbers with the method, you will eventually come up with 4,475X4,475 = 20.03. When you round down, the answer is 20.

Square of negative numbers

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Square negative numbers in the same way as above. Remember that a negative number multiplied by a negative number yields a positive number. So the square of a negative number is a positive number.

For example, -5X-5 =25. But 5X5=25. Remember that the square root of 25 can be -5 or 5. That means there will be 2 different square roots for a number.
Similarly, 3X3=9 and -3X-3=9, so the square root of 9 is both 3 and -3. Positive numbers are usually the main solution, so you only need to pay attention to this answer. ^{[6] X Research Sources}^{[7] X Research Sources}

Advice

You should remember the first few squares in the multiplication table:
- 0 ² = 0, 1 ² = 1, 3 ² = 9, 4 ² = 16.5 ² = 25.6 ² = 36.7, 7 ² = 49.8 ² = 64.9 ² = 81, 10 ² = 100,
- And also slowly memorize these spells: 11 ² = 121, 12 ² = 144, 13 ² 169, 14 ² = 196, 15 ² = 225, 16 ² = 256, 17 ² = 289…
- From the above simple squares, you apply the following squares: 10 ² = 100, 20 ² = 400, 30 ² = 900, 40 ² = 1600, 50 ² = 2500, …

Steps

Find the square root of an integer

Find the square root of other numbers

Square of negative numbers

Advice

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