Finding square root of ...

[fatbananas]

Can anyone tell me how one finds the square root of, say, a 3 digit number such as 169? DS did it by playing about with large numbers until he found that 13x13 got the desired answer. But is there a more efficient, time-saving way of figuring it out?

[aliportico]

If you know it's a whole number -

169>100 so your answer is more than 10.
169<400 so it's less than 20

Then think about the last digit - 9. What (one digit) number squared gives you something that ends in 9? 3*3=9, 7*7=49

So you've narrowed it down to 13 or 17. You could say that 169 is nearer to 100 than 400 so it's probably 13, or you could just work those two out to check.

[mystery]

Yes, that's a good way. We learned all our square numbers up to 25 x 25 off by heart. It was useful.

[aliportico]

I think for GCSE the suggestion is that you know 1-15 squared, and then 1, 2, 3, 4, 5, 10 cubed.

Knowing how to get from one square to the next e.g. knowing that 12x12=144 to 13x13 is useful: 144 + 12 + 13 = 169

[fatbananas]

Thanks Aliportico and Mystery. V useful advice.

[Tinkers]

I was taught a method for finding square roots when I was about 10. I can't remember it but it was a similar layout to long division.

I have never found anyone else who knows it to reteach me.

I'll go back to my slide rule now. (Yes I really do have one, freaks out the grads)

[southbucks3]

Knowing how to get from one square to the next e.g. knowing that 12x12=144 to 13x13 is useful: 144 + 12 + 13 = 169

I have just learnt this!

I asked my ds1 if he knew it , he Said he just knew his square roots to 20..then proved it!

I then asked him how it worked, and he said without even looking, it just does, who cares, if it works use it!

Told ds2, who is more his mother's son, and he immediately said "how"

I kind of see it of course, but would love a proper explanation please, to relay simply to ds2, as a blathered this morning like mums on the hoof do.

Tinkers, I bet my steel slide rule is older than yours, I have my dad's...it has 12th' s on the top line, with a smaller metric line below! It also has a grease line where he kept it sliding using something yukky and oily.

[russet]

I don't know if this is a proper mathematical proof, but if you get some graph paper and draw out a 12*12 square, to make it into a 13*13 square you have to add one length of 12 squares along the bottom edge and another length of 12 squares down the right hand edge. Then there is one more square to add in the corner where they meet.

Another way of working out non prime square roots is to see if you can divide the number. For example, suppose you had no idea the what the square root of 144 was. You could divide by 9 to get 144/9=16.
Therefore 144=9*16.
So square root of 144=(square root of 9)*(square root of 16)=3*4=12

[aliportico]

12x12=144
13x12=156 (i.e. add 12)
12x13=156
13x13=169 (i.e. add 13)

And I don't know the way you mention Tinkers, but I know it exists and I'd like to know it too!

[southbucks3]

Ok, it was the reverse I was leaving out, I will tackle it again with him tonight...thank you.