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When it comes to finding the square root of a number, our first instinct may be to reach for a calculator. However, there are techniques that allow us to find the square root without relying on digital assistance. This seemingly daunting task can be broken down into manageable steps, making it accessible to anyone willing to learn. By understanding the fundamentals behind mathematical operations and utilizing a variety of methods, we can uncover the square root of a number through mental math. This guide will explore these alternative approaches, equipping you with the tools to calculate square roots without the aid of a calculator.
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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
- 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
- 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.
- 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
- 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.
- 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
- 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
- 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] 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, …
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 without using a calculator may seem like a daunting task at first, but with the right understanding and practice, it is possible to do so. By utilizing various methods such as prime factorization, estimation, and long division, individuals can calculate square roots with relative ease. The process may require time and patience, but the satisfaction of being able to find the square root without relying on technology can be hugely rewarding. Moreover, practicing mental math and sharpening problem-solving skills is beneficial in various aspects of life. So, don’t shy away from this challenge, embrace it and soon you will be able to find square roots confidently without a calculator.
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