The square root of positive numbers eli5

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Explain to me like I’m 5, how to determine the square root of a random positive number. And I don’t mean common sense like 16=4, I need to understand the logic.

In: 6

You mean like how to solve it by hand? There are methods but they’re not easy. Back in the day before calculators, people would have used slide rules or tables for this sort of thing.

You can estimate square roots by looking at what square numbers your target number is between. E.g. if I wanted to know the square root of 42, I would observe that it is more than 36 and less than 49, so I know the square root must be between 6 and 7.

Either separating the factors, or long form division. Here’s copy paste from indeed:

Long-form factoring
Sometimes, you will not know what factors of a square root are squares. You can break the square root up into each individual factor, then solve it. For example, to long-form factor √225, here are the steps:
Find the factors. The most obvious factor of 225 is five, so you would start with √225 = √(5 x 45). You would further simplify by finding the factors of 45: √(5 x 5 x 9). The final factor you can simplify is 9, so your final long-form factor would look like √(5 x 5 x 3 x 3).
Pull duplicate factors out. When you see the same number twice as a factor, you will list it once outside of the square root symbol. In this case, we have two 5s and two 3s, so the equation will look like 5 x 3.
Solve the remaining equation. The last step is to finish solving the equation. In this case 5 x 3 = 15.

Long division method
There are cases where you might not recognize the factors immediately. The long division method lets you find the square root without having to estimate. For this method, we will find √361. Here are the steps to the long division method:
Separate your square root base into pairs. Starting from the right, group the number into pairs. In our example, 361 would be 3 61.
Find the largest square that divides into the first number or pair. This will give you the first number in your answer. The first number on the left is 3. The highest square that goes into it is one because 1 x 1 = 1, while 2 x 2 = 4.
Subtract the square from the first number or pair. Subtracting the square from the first number will give you a remainder, which will be included in the next step. In this example, 3 – 1 = 2.
Drop down the next pair. The next number you will work with will be the combination of the subtracted square and the next pair. In this case, they would make a three-digit number. When you bring 61 down, the number you will use for finding the next digit in the square root is 261.
Multiply the first digit of the square by two. This will be the first digit in the factor for finding the second digit of the square root. In the example, the first digit of the square root is one. 1 x 2 = 2.
Set up the next factor equation. The equation for the next step is based on the digit from step five and the number from step four. The first factor will be a two-digit number, where the first digit is the number from step five. The equation will look like 2_ x _.
Find the number that fills the blanks. This number will be the next digit in the solution for √361. The number that will fill the blanks will be the same, and it will be the highest digit where the factors are less than or equal to the number in step four. In this example, the goal number is 261. We will start with 9, so the equation will look like 29 x 9 = 261.
Put the number next to the first digit. In this example, the square is 19.

I’m going to explain like you’re 11 because 5 year olds don’t need square roots.

The square toot of any number is what you have to times by itself (like 6 x 6 or 27×27) to get your “square root of number”

What do you have to times by itself to get 25? 5. So the square root of 25 is 5.

Just look at these times tables by themselves:

1×1 = 1
2×2 = 4
3×3 = 9
4×4 = 16
5×5 = 25
6×6 = 36
7×7 = 49
Etc. Just look at it backwards and that’s your square root. The quare rott of 36 is 6.

It gets funny when you need the square root of something that doesn’t fit nicely. The square root of 5 is going to be between 2 and 3 because 2×2 is 4 and 3×3 is 9, and 5 is in between. 5 is closer to 4 than 9, so you can guestimate something closer to 2 like 2.2 (instead of guessing some long like 2.7 or even 2.5) as the square root. To check, you times it by itself and see if you need to increase or decrease your guess. 2.2x 2.2 is 2.84 (calculator is fine), so guess 2.3 (2.3×2.3=5.29) which is too high. This means the square root of 5 is between 2.2 and 2.3. keep playing a shrinking game of money in the middle with it untill you’re happy with your accuracy. A lot of square roots are numbers that don’t really end.

There are many ways to find an approximation of a square root. One of the oldest known and easiest to understand is the Babylonian method. If you are trying to find the square root of N, start with some guess x. Then take the average of x and N/x and that then becomes your new value for x. Repeat those steps until the difference between x and N/x is small enough that you’re satisfied with the level of precision. This works because if x is too small, N/x will be too large, and if x is too large then N/x will be too small. The true value is always somewhere in between. It will actually work with any starting value of x, but it will converge on the true value faster if you start with a good approximation.

Outside of those “simple” roots you can NOT determine a proper square root, you can only approximate it.

That’s because square roots are either integers if the number is a perfect square (that’s the “common sense”) or irrational, for any other number.

Irrational numbers have infinite non-repeating representations as decimals (or any other numeric system).

The ways of calculating the roots are so complicated because they aren’t really calculating it directly, only finding out more and more digits of the answer in an infinite loop.

Well you might want the proper definition of powers. a^b = c^[b×log_c(a)]. Since c could be anything we can use the exp function. a^b = e^[b×ln(a)]. A square root would be b=0.5. Of course computing the values of the exp function and the ln function is also difficult to do by hand, but this is the general form of powers. a and b could be any real number and this formula will work.