I Can Now Do Square Roots and Cube Roots By Hand

This isn't necessarily a question to be answered. But more like "lost knowledge" that I can now do. Of all of my grades K-12 and college courses up to Calculus, not a single teacher or professor taught anyone how to do square roots and cube roots or even nth roots by hand. Although nth roots are tricky starting at the 4th root, you can cheat it by doing a double square root. Even my textbooks where showing how to do it in a calculator. And I feel that a calculator shouldn't be dependent on forever. Because technology will ultimately dumb us down if taken too far. For now, I will show you how to do square roots and cube roots by hand.

Lets start with the square root of 144 for example. Something simple.

Code:

---

  _____
\|144

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Step 1) Underline the numbers in double digits beginning from the right to the left. If there is one digit left, then underline that one digit.

Code:

---

_____
|144
 - --

---

Step 2) Next, ask yourself this. What two numbers of the same number multiplied together will be or get as close to 1 as possible without overshooting it?

1² = 1
So the answer is 1.


Put a 1 over the square root. And subtract the 1 digit from the 144 by 1. Then drop the next 2 digits. If no digits are available and its not zero after the subtraction, drop 2 0's and add a decimal.

Code:

---

_1___
|144
-1 vv
-----
 044

---

Step 3) Since you got 1², instead of multiplying by 1*1, add 1+1 to get 2, but leave a blank space after 2 to get 2_.

Code:

---

    _1___
2_|144
    -1 vv
    ----
    044

---

Step 4) Now ask yourself. What 2_ times its own missing digit will be or get as close to 44 as possible without overshooting it?

20 * 0 = 0
21 * 1 = 21
[22 * 2 = 44]

Step 5) Since you got 2, put it on top, and subtract it by 44.

Code:

---

    _12__
22|144
    -1 vv
    -----
    044
    - 44
    -----
        0

---

And that is it! Now if it were a bigger number or a decimal, you would keep adding blank digits. Like if we had to keep going, we would drop 2 0's, add 22 + 2 to make 24, and put a blank next to 24_. Now ask yourself. What 24_ times its own missing digit will be or get as close to 0 as possible without overshooting it? 0! We can't go no further. If you really wanna test this technique, try any number. Hell try even the square root of 2. It'll never end, but rest assure the number will match in your calculator ;)

Now on to cube roots. This is a little tricky, as there are a few more steps involved. But nothing we can't handle. Lets try the cube root of 91125

Code:

---

3 _____
\|91125

---

I used this number so you can see the technique first hand.

Step 1) Underline the numbers in triple digits beginning from the right to the left. If there is one digit left, then underline that one digit. If there are two, underline the two digits.

Code:

---

______
|91125
  -- ---

---

Step 2) Next, ask yourself this. What three numbers of the same number multiplied together will be or get as close to 91 as possible without overshooting it?

1³ = 1
2³ = 8
3³ = 27
[4³ = 64]
5³ = 125

So the answer is 4.

Put a 4 over the cube root. And subtract the 91 digits from 91125 by 64. Then drop the next 3 digits. If no digits are available and its not zero after the subtraction, drop 3 0's and add a decimal.

Code:

---

_4___
|91125
-64 vvv
-----
 27125

---

Step 3) Since you got 4³, instead of multiplying by 4*4*4, add 4+4+4 to get 12, but leave a blank space after 12 to get 12_.

Code:

---

      _4___
12_|91125
    -64 vvv
      -----
      27125

---

Step 4) Now multiplying the 12_ by its own digit 3 times will not work necessarily. There is another trick involved here. Were gonna involve the first 3 digits of our final number, which is 271, and divide it by the 12 we obtained times the 4 we obtained to get an approximation rounded down. We only need the whole number from this:

271/(12*4) ≈ 5


Step 5) With this number you have obtained, you will multiply it by the 4 you obtained earlier squared, multiplied by 300.

5 * 4² * 300 = 24000

Step 6) Plug in the missing value for 12_ with 5 and multiply by 5 twice.

125 * 5 * 5 = 3125

Step 7) Add em up

24000 + 3125 = 27125

Step 8) Subtract it from the total, and put the 5 above the cube root. It became 0. Thats it!

Code:

---

      _45__
125|91125
    -64 vvv
      -----
      27125
    -27125
    --------
            0

---

The final answer is 45. I honestly don't understand why they don't teach this in elementary or middle school cause it was so damn easy to figure out. And the technique is dying due to a high abundance of calculators. Even the textbooks are telling you to use a calculator! Let me know what you guys thing. :bigyello:

I think you answered your own question by outlining 8 manual steps, tricky rules, and numerous calculations to do something that can be done with like 7 or 8 keystrokes on a calculator.

It wasnt a question though like I said. The problem is, is that what if computers and calculators no longer exist due to a catastraphe? How would you calculate then? That is the knowledge that is becoming lost due to technology, and is not being taught in schools ;)

Quote:

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Originally Posted by Jacob Roman

It wasnt a question though like I said.

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I was replying to this:

Quote:

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Originally Posted by Jacob Roman

I honestly don't understand why they don't teach this in elementary or middle school cause it was so damn easy to figure out.

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And I'll posit that they ARE teaching the method you describe above, and I can outline why.

Given a square number, call it S

S can be represented as (a+b)^2

Expanding (a+b)^2 we get a^2 + 2*a*b + b^2

Now, I will posit that, in the case of a square root, your first two steps above are finding the value of a. You are finding it as a single digit, but the value of a is simply the value you find * 10^u, where u = the number of double-underlined sections to the right of the left-most digit (or two digits in the case of a number that is an even number of digits).

So, you found 1, but you are really finding that a = 10.

Now, you are subtracting a^2, or 10^2, or 100. Thus, given S = (a+b)^2 = a^2 + 2*a*b + b^2, you are subtracting a^2 and we are left with 2*a*b + b^2

Now, you say to take the initial digit that you found and multiply by two and put an empty space next to it. That is equivalent to multiplying a by 2. Inside the "empty space" we find the remaining digit that gives us the answer to the square root. Well, since we know that S = (a+b)^2, and we've found a, that remaining digit represents b.

So lets go back to our equation. We're left with 2*a*b + b^2, and we are now dividing by 2*a + _ (blank), but we know the blank = b, so we are dividing by (2*a+b)

So (2*a*b + b^2)/(2*a+b) = b*(2*a+b)/(2*a+b) = b, and that is what you are finding in step 5.

So performing the steps you are performing, you are using a technique to turn a square number S into (a^2) + (2*a*b + b^2) = (a+b)^2 which is, in fact, being taught in school as a simple binomial expansion.

Quote:

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Originally Posted by Jacob Roman

I honestly don't understand why they don't teach this in elementary or middle school ...

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AFAIR, we had that in school (germany)...

It's trivial and also mentioned in the english Wikipedia-section (among dozens of other manual methods):
https://en.wikipedia.org/wiki/Method...g_square_roots

Olaf

I was taught that in my maths class when I was 13. We even had it as an exam question. No calculators back in the early 1970's though!

Now teach yourself to use a slide rule. After all, maybe calculators will go away??

(I gotta say that surfing the web on a slide rule is pretty tough)

My basic point isn't so flippant, though. We lose LOADS of knowledge with all kinds of advances. Do all of those things matter? For example, navigation isn't a totally lost art, but it's likely that the majority of people can't calculate the magnetic declination for their area, nor navigate successfully using a map and compass. Those are skills more likely to matter than being able to calculate a square root by hand, but they are also often left out of school (not mine, but many schools don't cover that). Another area would be knots, which were once essential to lots of trades and often aren't covered all that well anymore.

The simple fact is: None of us knows more than a small fraction of all the knowledge of humanity by now. There's always more to learn, and more to learn than we have time to learn. So, you'll have to decide for yourself what to leave in and what to leave out.

Quote:

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Now teach yourself to use a slide rule. After all, maybe calculators will go away??

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Slide rule?? Use good old fashioned log tables for calculations like we had to do in the 1970's! We didn't get slide rules until our fourth senior year. We could then use them in exams but lost marks if you weren't accurate. Even at A level, all we had were log tables and a slide rule.

Times change, and just as there's now no justification for learning to undertake calculations via log tables, there's no justification for learning how to do manual square roots etc. There's other more important maths that can be learnt instead. There is only a fixed number of school hours after all.

Our tables were wood, too, but not logs exactly. Logs were for heating.

Still have my father's slide rule, though.

Quote:

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Originally Posted by Jacob Roman

It wasnt a question though like I said. The problem is, is that what if computers and calculators no longer exist due to a catastraphe? How would you calculate then? That is the knowledge that is becoming lost due to technology, and is not being taught in schools ;)

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I can't figure out what catastrophe would cause the loss of computers and calculators, without more important needs being affected.

If such a catastrophe occurred, then it seems like the ramifications would be much wider than the need to figure out how to do square roots or cubes.
I would think survival matters, like farming and hunting and possibly defending oneself might be much higher on the needs list at that point.