Hathaway

Being Mental - (in calculation) Squaring Numbers

What follows is a motley and pretty random collection of techniques or shortcuts, collected over many years, to help speed up calculations in your head. They are not in any particular order, and I will not pretend that all of these are the full toolbox by any means, but they are probably the most accessible ones. You will be spared all of the algebraic stuff supporting what follows, but I will be happy to supply the same to anyone that wants it.

Numbers ending in 5

Consider the number 35. To square this:

  • Take the first number 3 and multiply by the next number i.e. 3+1, which of course is 4
  • Taking the answer 12, simply pop 25 on the end to give 1225

This works for any number ending in 5 : for example. 135 squared is 13 x14 with 25 popped on the end, which is 18225

 Numbers near to 50

Consider the number 54. To square this:

  • Deduct 25 from 54 to give 29. Pop 00 on the end to give 2900
  • Add the square of the difference between 50 and 54 which of course is 4 x 4
  • So the answer is 2900 plus 16 = 2916

Consider the number 43. To square this:

  • Deduct 25 from 43 to give 18. Pop 00 on the end to give 1800
  • Add the square of the difference between 50 and 43 which is of course 7 x 7
  • So the answer is 1849

Numbers near to, but less than 100

Consider the number 98. To square this:

  • Double the 98 to 196, then discard the 1 and stick 00 on the end to give 9600.
  • Subtract 98 from 100 to give 2, which you square and add to the end
  • So the answer 9600 + 4 = 9604

Consider the number 87. To square this:

  • Double 87 to 174, then discard the 1 and stick 00 on the end to give 7400.
  • Then subtract 87 from 100 to give 13, which you square and add to the end
  • So the answer 7400 + 169 = 7569
Numbers greater than 100 but less than 150

Consider the number 112. To square this:

  • Double the last two digits to give 124, then pop 00 on the end to give 12400. 
  • To this add the square of the difference between 112 and 100 which is 12 x 12 = 144. 
  • Add this to 12400 to give the answer 12544

Consider the number 132. To square this:

  • Double the last two digits to give 164, then pop 00 on the end to give 16400. 
  • To this add the square of the difference between 132 and 100 which is 32 x 32 = 1024. 
  • Add this to 16400 to give the answer 17424.
For other numbers that are near the hundred breaks (200, 300,…..up to 1000) 

For these, I turn to the classic quadratic model. Two examples follow:

Consider the number 989. To square this:

  • Think of  989 in terms of (1000 – 11) x (1000 – 11), OR 
  • More easily, think of it as 1000 squared [which is 1000000] minus 2 lots of 1000 times 11 [22000] giving 978000. 
  • Then add the 11 squared [121] to give 978121

Consider the number 691. To square this:

  • Think of 691 in terms of (700 – 9) x (700 – 9), OR 
  • More easily, think of it as 700 squared [which is 490000] minus 2 lots of 700 times 9 [12600] giving 477400. 
  • Then add the 9 squared [81] to give 477481
For other numbers that are near the 50 breaks (250, 350,…..up to 950) 

the quadratic model also suffices. Two examples follow:

Consider the number 367. To square this:

  • Think of it in terms of (350 + 17) x (350 + 17). 
  • Because of what we now know about numbers ending in 5 (see above), we know 350 squared is 122500. 
  • To this we need to add two lots of 350 [700] times 17 which is 11900 to give 134400 
  • To this add 17 squared [289] giving the total 134689

Consider the number 637. To square this:

  • Think of it in terms of (650 – 13) x (650 – 13). 
  • Because of what we now know about numbers ending in 5 (see above), we know 650 squared is 422500.
  • To this we need to subtract two lots of 650 [1300] times 13 which is 16900 to give 405600. 
  • To add 13 squared [169] giving the total 405769.

This looks all a bit daunting written down, but when you start kicking this around in your head the logic flows a lot easier for not having to move your eyes over marks you make on paper. Start with easier numbers (near 100 and 50 break points) and move onto more complex numbers.

Where do you go from here then? Well, this can be extended to larger numbers by applying some of the techniques in the multiplication page. I’ll leave you to think about this – happy squaring!