Let's say we wanted to check if 16 was a perfect square or not.
What we do is subtract off the list of odd numbers {1,3,5,7,9,...}. If we reach 0 at any point, then the step number is what the square root will be.
So we'll start with 16 and subtract off 1 to get 15. Then from 15, we subtract off 3 to get 12. From 12, we subtract off 5 to get 7. This process is laid out below
- 16-1 = 15
- 15-3 = 12
- 12-5 = 7
- 7-7 = 0
By step 4, we reach 0. This indicates that [tex]\sqrt{16} = 4[/tex]
Note: The numbers to the left of the decimal point indicate the step number. All values mentioned are whole numbers.
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Here's another example. I'll start with 49
- 49 - 1 = 48
- 48 - 3 = 45
- 45 - 5 = 40
- 40 - 7 = 33
- 33 - 9 = 24
- 24 - 11 = 13
- 13 - 13 = 0
We reach 0 at step 7, therefore [tex]\sqrt{49} = 7[/tex]
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If we started with some non-perfect square, say 50, then we'd get this:
- 50 - 1 = 49
- 49 - 3 = 46
- 46 - 5 = 41
- 41 - 7 = 34
- 34 - 9 = 25
- 25 - 11 = 14
- 14 - 13 = 1
- 1 - 15 = -14
As you can see, we don't reach 0, which means 50 is not a perfect square. The closest we get is 1 and that happens on the 7th step. This suggests [tex]\sqrt{50}[/tex] is closest to 7. It turns out that [tex]\sqrt{50} \approx 7.071[/tex] which helps confirm that previous statement.
