Finding Square Root, √ , Without a Calculator

[joanandkuharajasekaram]

@Mr H: On the last example, instead of 100-97, you could use 97-100 (as same as in the first two examples) and this would lead to -3 on the numerator. So the method works the same way for the upper as well as the lower approach.

[incendiohawk]

This can be generalised into nth root of (x^n + a)^(1/n) ≈ x + a/(nx^(n-1))
e.g. cube root of 70 = (4^3+6)^(1/3) ≈ 4 + 6/((3)(4²)) = 4.125. The actual value is 4.121 to 3 decimal places

[ockham]

Brilliant teaching. Succinct and very clear.

[smalin]

I'd like to see a geometric explanation of why this approximation works.

[hywell_]

Just learned linear approximation recently so it finally makes a bit more sense to me now why this method works.

[MeMadeIt]

Thx! 👍 I had long forgotten how to do this. 😄
Since school, I've mostly just guesstimated the difference between the radicand and its closest perfect squares. 😄

[-.-]

Cool, I haven’t done square roots since school in the early 1970”s.😊 and we didn’t have calculators just paper, pencil, and slide rule. I didn’t get a calculator til I was in college. Part of me wants to learn advanced math again. Now that I’m older, things make more sense 😂 I’ve subscribed.

[dogslife]

Mr H's teaching method is very polished
Good mixture of technology

[StereoSpace]

Awesome teaching, as always with Mr H.

[nafnist]

This is not finding the squareroot, but doing an approximation.

[Phylaetra]

It is important to keep in mind - you are not actually finding the square root - you are approximating the square root. The approximation it the worst what the number is between two squares; for example 156 is 12 more than 144 or 13 less than 169, giving you approximations of 12.500 either way; but the actual root is 12.490.

There are two methods that you can use to calculate a square root - one is Newton's method (using a method of his from Calculus, though the actual method was known in Babylonian times) and another is called the 'long-division method', so named because it looks kind of like long division.

Newton's method is very fast, and pretty easy to do. Guess what the root of your number n is (say r_1), then find the average of r_1 and n/r_1; call this number r_2. Repeat - find the average of r_2 and n/r_2. Continue until you are as close as you need to be.

For 150, we can start with 12: average 12 and 150/12:
1/2 * (12 + 150/12)
= 1/2 * (144/12 + 150/12)
= 1/2 * (294/12)
= 147/12 = 12.250
But if we do that again we get:
43209/3527 = 12.24744898, the actual value being 12.24744871 -- off in only the last two decimal places!

You need to work it out as fractions - but you double the number of accurate decimal places with each time you do it.

The 'long division' method is a bit more difficult to show in a comment like this, but it has the advantage of stopping if you have a perfect square, and everything stays as a decimal. It adds one decimal point with each row, basically, but - each additional decimal point takes a little more work.

[robhill]

He is using the derivative of X^0.5 to find the fractional part added to the perfect square. Easy!

[CommDao]

I'm not smart, but this made me feel like I have the potential to be smart. 🙏

[bowlineobama]

I have never seen this method before, but you still have to deal with fractions which you may still have to use a calculator to take care of the fraction portion. I recommend using the binomial expansion method.

[charlespartrick]

Fantastic - another great method I can teach my students.

[whatzause]

Very good and will be occasionally useful. I enjoyed your presentation. What kind of audio system are you using? I found the audio somewhat muffled due to an apparent deficiency in the high frequency range. Therefore, it would have been better if I could have understood what you said better.

[kingbeauregard]

Here is a reason to do this even if you have a calculator: if you are, for whatever reason, trying to find the difference between sqrt(64) and Your calculator may have trouble calculating that, or at least displaying it with enough precision.

[rogerphelps]

This is just Newton's method for successive approximaion of a square root.

[wilmenmedina]

Shame on you for those who say “still need calculator for fraction” 😮

[claireli]

This method is derived from the approximation of the first two terms of the binomial expansion (a+x)½ where a is the required perfect square.