The X+Y Files

Issue 7

Approximately Square

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It isn’t too difficult to see that 

68  is between 8 and 9, since 8² = 64
and 9² = 81, but a closer approximation can be obtained without knowing the square numbers at all – OK, so you probably know your square numbers up to 10² = 100, or perhaps up to 20² = 400 [I think you should! – Ed.], but I doubt if many people know 167² off the top of their heads!

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To find the approximate value of

68 we start by subtracting 
the odd numbers in order, starting from 1, until we can go no further.
As you can see, it is possible to subtract the first 8 odd numbers before getting stuck, and leaving a ‘remainder’ of 4.

The ‘answer’ is then approximately 8 4/16 

The ‘8’ is the number of odd numbers subtracted.

The numerator of the fraction is the remainder

The denominator is one more than the last number subtracted

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So we have an approximate value for  

68  as 8.25,
or 8.3 to one decimal place.

In fact 8.25² = 68.025 and 8.3² = 68.89.

 

68
 -1
 67
 -3
 64
 -5
 59
 -7
 52
 -9
 43
-11
  32
-13
  19
-15
   4

So how does this work? Firstly, adding up the odd numbers from 1 gives the square numbers

(1 = 1²;  1 + 3 = 4 = 2²;  1 + 3 + 5 = 9 = 3²;  1 + 3 + 5 + 7 = 16 = 4²;  …)


so in subtracting 8 odd numbers (but not being able to subtract any more) we know that
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8 <  

68  < 9.

That leaves us with 4, but how do we use it, and where does the 16 come from?

Think about multiplying out
(8 + 4/16)²  =    (8 + 4/16)(8 + 4/16)  =   8² + 2 × 8 × 4/16 + (4/16)²

16, being one more than the 8th odd number, is the eighth even number, and will cancel with the 2 × 8 in the middle term, just leaving our ‘remainder’, 4. The ‘error’ term, which we are neglecting, is (4/16)² and in this case is very small, but will always, of course, be less than one.

Mike Moon