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Finding the square root of a number is a fundamental mathematical operation that is often used in various fields such as engineering, physics, and computer science. While calculators make this task effortless, there are situations where access to a calculator is limited or not possible. In such cases, being able to find the square root without using a calculator becomes a valuable skill. By understanding the principles and techniques underlying the process, individuals can develop methods for finding square roots manually. In this article, we will explore different methods and strategies that can be employed to find the square root of a number without relying on a calculator, allowing for a greater understanding and fluency in mathematical computation.
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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.
Steps
Find the square root of an integer
![Image titled Find a Square Root Without a Calculator Step 1](https://www.wikihow.com/images_en/thumb/1/11/Find-a-Square-Root-Without-a-Calculator-Step-1-Version-3.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-1-Version-3.jpg)
- 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] XResearch Source
![Image titled Find a Square Root Without a Calculator Step 2](https://www.wikihow.com/images_en/thumb/1/1d/Find-a-Square-Root-Without-a-Calculator-Step-2-Version-3.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-2-Version-3.jpg)
- 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.
![Image titled Find a Square Root Without a Calculator Step 3](https://www.wikihow.com/images_en/thumb/b/bd/Find-a-Square-Root-Without-a-Calculator-Step-3-Version-3.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-3-Version-3.jpg)
- N is the number for which you are looking for the square root. N will be in the radical sign. [3] XResearch 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
![Image titled Find a Square Root Without a Calculator Step 4](https://www.wikihow.com/images_en/thumb/0/02/Find-a-Square-Root-Without-a-Calculator-Step-4-Version-3.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-4-Version-3.jpg)
- 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] XResearch 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.
![Image titled Find a Square Root Without a Calculator Step 5](https://www.wikihow.com/images_en/thumb/e/ed/Find-a-Square-Root-Without-a-Calculator-Step-5-Version-3.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-5-Version-3.jpg)
- Then divide the number you have by those square roots. Find the average of the quotient you just found with the corresponding divisor (the average is the sum of two numbers divided by two). Then divide your divisor by the average. Then find the average of the answers you just split.
- Sounds troublesome right? Here is an illustrative example. For example, 10 is between the two perfect squares 9 (3X3=9) and 16 (4X4=16). The square roots of these two numbers are 3 and 4. Divide 10 by the first number, 3. You get 3.33. Now, you find the average of 3 and 3.33 by finding their sum and then dividing the sum by 2. You get 3.1667. Now divide by 10 by 3.1667. The answer is 3.1579. Find the average of 3.1579 and 3.1667 by adding them together and dividing the sum by 2. Your final answer is 3.1623.
- Double-check your answer by multiplying your answer (in this case, 3.1623) by itself. 3.1623 times 3.1623 equals 10,001.
Square of negative numbers
![Image titled Find a Square Root Without a Calculator Step 6](https://www.wikihow.com/images_en/thumb/9/9e/Find-a-Square-Root-Without-a-Calculator-Step-6-Version-3.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-6-Version-3.jpg)
- 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] XResearch Sources[7] XResearch Sources
![Image titled Find a Square Root Without a Calculator Step 7](https://www.wikihow.com/images_en/thumb/5/53/Find-a-Square-Root-Without-a-Calculator-Step-7.jpg/v4-460px-Find-a-Square-Root-Without-a-Calculator-Step-7.jpg)
- Find the alignment mark on the calculator.
- The online calculator will ask you to enter the number you want to find the square root of and press a button. The calculator will then find the square root of that number. [8] XResearch 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, …
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 34 people, some of whom are anonymous, have edited and improved the article over time.
There are 8 references cited in this article that you can see at the bottom of the page.
This article has been viewed 136,700 times.
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.
In conclusion, finding the square root of a number without using a calculator is an achievable task that requires patience and understanding of mathematical principles. While it may seem daunting at first, using methods such as prime factorization, estimation, and the Babylonian method can help simplify the process. By breaking down the problem into smaller steps and utilizing these techniques, one can confidently find the square root of a number by hand. Additionally, practicing mental math and becoming familiar with perfect square numbers can further enhance the ability to quickly approximate square roots. Ultimately, developing these skills not only strengthens one’s mathematical prowess but also enhances problem-solving abilities and critical thinking skills. So, the next time you find yourself without a calculator, remember that with a little practice and determination, you can still solve for the square root of any number by hand.
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