## Interview Answer

9 Answers

If you can do a little multiplication and adding in your head this is not too difficult.

First get a rough estimate about where you need to be.

Try 40 x 40 = 1600. Good but you can do better.

Try 44 x 44 = (44x40) + (44x4) = 1760 + 176 = 1936.

Now try 45 x 44. All this is saying is we need 45 sets of 44. So take 1936 and add 44 to it. The result is 1980.

Now try 45 x 45. All this is saying is that we need 45 sets of 45. Since we can reverse numbers in multiplication we already have 44 sets of 45 which is 1980. Just add 45 to 1980 and the result is 2025.

So we know the square root is somewhere between 44 and 45. The range from 1936 to 2025 is 89. 2000 - 1936 = 64. So what is 64/89? Round it up so we get a fraction like 70/100 which is .7.

Therefore, 44.7 is a good estimate. The actual root is 44.72135954....

Hope that helps.

If you are rusty with numbers in your head then I would be all means follow Mike's way.

Haven't done math like that in my head since college but I'd start off at 40x40 = 1600. the diff between x^2 and (x+1)^2 is 2x +1. so I'd go to 1681 (+83), 1764 (+85), 1849 (+87), 1936. then extrapolate linearly between 44^2 and 45^2.

I got a little lucky with the numbers. I know that 20 * 20 = 400 * 5 = 2000. I also remembered that 22 * 22 = 484, so 2.2 * 2.2 = 4.84 which is close to 5. So I had somewhere between 44 and 45. If I had a different number, like 1377, I wouldn't go more than the whole number * the square root of the remainder.

an irrational number

You can easily multiply a number ending in 5 by itself, as follows:

35 * 35; take the first digit (3), now add one to make (4), multiply (12), follow by 25, = 1225

45 * 45: so, as above, (4), times (4+1), = 20, follow by 25, so = 2025

Since 2025 is greater than 2000 by only 25 (about 1/4 of 45 + 45), I would guess the square root to be 45 - 0.25, or about 44.75

Decompose to prime number powers:

2000=2*1000=2 * 10^3 =2^4 * 5^3

Apply 1/2 power to compute sqrt:

sqrt(2000) = 2^(4/2) * 5*(3/2)

sqrt(2000) = 2^2 * 5 sqrt(5)

sqrt(2000) = 20 sqrt(5).

Evaluate sqrt(5) - an irrational number indeed:

5=1+2^2

Sqrt(5) > 2

Ballpark: Since we're off by 1, try a small number:

2.1*2.1 = 4.41

2.2*2.2= 4.84

2.3*2.3=5.29

20 * 2.3 > 20 sqrt(5) > 20 * 2.2

46 > 20 sqrt(5) > 44

Not precise enough yet...

To have more precision, John says:

45^2 - 44^2 = 89

2000 - 44^2 = 64

so:

sqrt(2000) = 44 + 64/89

[I am dubious of that being a rule that works.. Let's check]

s^2=a^2 + (s^2-a^2)/(s^2-(a+1)^2)

s=a2+(s2-a2)/(s2-a2-1-2sa)

s * (s2-a2-1-2sa)=a2 * (s2-a2-1-2sa)+(s2-a2)

s4 -s2a2 -s2 -2s3a = a2 s2 -a4 - a2 - 2sa3 + s2 - a2

0 = s4 - s2a2 -s2 -2s3a - a2 s2 + a4 + a2 + 2sa3 - s2 + a2

0 = s4 + a4 - 2 s2 (a2-1) + 2sa3 -2s3a + 2 a2

Argh!

Anyway, back to approximating by hand:

6/9 =2/3=0.66

0.4/8=0.05

Result: 44.71

The square root of 2000 is a number that when multiplied by itself (once) is equal 2000

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sqrt(2000) = sqrt(100 x 20) = 10 sqrt(20).

sqrt(20) = sqrt(4 x 5) = 2 sqrt(5)

I wouldn't be able to guess sqrt(5) very accurately other than it is greater than 2 and less than 3.

sqrt(2000) = 44.7 (calculator), though 20 sqrt(5) would probably be good enough.

MikeonDec 30, 2010