You can even calculate square root of pi on your own
These days the vast majority of learners are used to relying on calculators when it comes to finding square root of pi or any other math stuff. However, frankly speaking, that’s true not only for students, but also for experienced professors. People simply forgot that not so long ago, in the 20th century they were capable of calculating square root of pi or anything else manually. God only knows, but maybe someday there might be something wrong with your calculator. We’re just dropping a hint that any person should know how to carry out calculations without any electronic hardware to back up its probable absence for example.
Calculating square roots by hand
In fact, there’re several effective methods to calculate square roots manually. One of them suggests the use of prime factorization. So, let’s get familiar with the very essence of this math technique.
In simple words, you require dividing your number into perfect square factors. The given method makes use of a number’s factors in order to calculate a number’s square root. As for a number’s factors, they can be defined as any set of other numbers to be multiplied together to make it. For example, two and four appear to be the factors of eight, because two multiplied by four is eight. Perfect squares are simply whole numbers, which appear to be the product of other whole numbers. 25, 36, and 49 can be associated with perfect squares because they’re respectively 52, 62, and 72. As you might have already guessed, perfect square factors are those factors, which are perfect squares. In order to calculate a square root by means of prime factorization, you require reducing your number into its perfect square factors.
Let’s make use of a persuasive example. For instance, you’d like to calculate the square root of 400 manually. First of all, you need to divide the assigned number into perfect square factors. Taking into account that 400 appears to be a multiple of 100, we know for sure that it’s divisible by 25, which is a perfect square. Via mental division, we find out that 25 goes into 400 up to 16 times – also a perfect square. Therefore, the perfect factors of 400 appear to be 25 are 16. That’s because 25 multiplied by 16 is 400. This could be written as Sqrt(400) = Sqrt(25 × 16).
The product property of square roots suggests that for any given numbers b and a, Sqrt(a × b) = Sqrt(a) × Sqrt(b). Thanks to this property, you have an excellent opportunity to take the square roots of the given perfect square factors and then multiply them together to obtain the long-awaited answer.
As for our example, Sqrt(25 × 16) or 5 × 4 = 20.
Keep reading this review to explore charming mysteries of the world of mathematics, including square root of pi and many more.
If your number can’t be factored perfectly, it makes sense to reduce your answer to simpler terms. By the way, in real life in most cases, the numbers you’ll require finding square roots for can hardly be nice round numbers with apparent perfect squares factors such as 400. Respectively, it might not be possible to calculate the precise answer as a integer. Nevertheless, by simply finding any perfect square factors you’ll be able to get the answer in terms of a simpler, smaller and easy-to-manage square root.
In order to do this, you require reducing your number to a combination of perfect square factors as well as non-perfect square factors. Then, you should simplify the newly obtained stuff.
Ok, let’s employ the square root of 147, for instance. We see that this number isn’t the product of two perfect squares, so one can’t obtain an exact integer values as mentioned above. Nevertheless, that’s the product of a perfect square and another number. These are 49 and 3.
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- Sqrt(147)
- = Sqrt(49 × 3)
- = Sqrt(49) × Sqrt(3)
- =7 × Sqrt(3)
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Estimate, if required. With your square root, it’s not difficult to make a rough estimate of a numerical answer by simply guessing the value of any remaining square roots and then multiplying this stuff. In fact, there’s one way to guide your estimate. You need to calculate the perfect square on either side of the number of your square root. The decimal value of the number of your square root will be between these numbers.
Let’s get back to our example. Considering that 22= 4 and 12= 1, we’re already aware of the fact that Sqrt(3) is somewhere between 1 and 2 and most likely closer to 2 than to 1. So, 1.7. 7 × 1.7 =11.9. Just check this newly obtained result in your calculator and you’ll see with your own eyes that the given outcome is quite close to 12.13 – the actual answer.
The given approach works for larger numbers too. For instance, you can estimate Sqrt(35 between 5 and 6. Well, 52= 25 and 62= 36. You see that 35 appears to be between 25 and 36, therefore its square root will most likely be between five and six. Taking into account that 35 appears to be just one way from a larger number, 36, one can conclude that its square root is lower than six. Just check this stuff with a calculator. The real result will be 5.92. As you see, your manual calculation has brought a very close outcome. Don’t overlook square root of pi in this guide.
Reducing your number to its lowest common factors at the first step would be an alternative solution. So, you don’t need to find perfect square factors if there’s a possibility of determining a number’s prime factors or simply prime numbers. In this particular case, you require writing your number out in terms of its lowest common factors. After this, you need to seek matching pairs of prime numbers among your factors. Once you spot two prime factors, matching each other, exclude them from the square root and then put one of these numbers outside the square root.
For instance, let’s calculate the square root of 45 with the help of this technique. So, we already know that 45 = 9 × 5 and 9 = 3 × 3. You can write your square root in terms of its factors. For instance, Sqrt(3 × 3 × 5). Just remove the 3’s and place on 3 outside the square root in order to obtain your square too in simple terms. That’s (3)Sqrt(5).
You’re definitely making decent progress as for square root of pi and other related stuff. Now let’s have a closer look at another example. We encourage you to calculate the square root of 88:
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- Sqrt(88)
- = Sqrt(2 × 44)
- = Sqrt(2 × 4 × 11)
- = Sqrt(2 × 2 × 2 × 11). Ok, we’ve got several 2’s in this square root. Considering that two appears to be a prime number, we have the right to remove a pair and place one outside the square root.
- In simple terms our square root would be (2) Sqrt(2 × 11) orsimply (2) Sqrt(2) Sqrt(11).Respectively, you can easily estimate Sqrt(2) as well as Sqrt(11) and then calculate an approximate answer.
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Making use of a long division algorithm
The method suggests separating the digits of your number into pairs. First of all, you should draw a vertical line to separate your work area into two sections. After this, you should make a shorter horizontal line close to the top of the right section in order to divide the right section into a tiny upper section and a larger lower section. Then, you require separating the digits of your number into corresponding pairs. Just start from the decimal point. Go on reading, you’ll get closer to square root of pi.
For instance, you’ve been assigned to calculate the square root of the number 780.14. Here you’re expected to draw two lines in order to divide your workspace and write 7 80. 14 at the top of your left space.
The given approach enables you to calculate the largest integer n with square lesser than or even equal to your leftmost number.
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