Before this page: be able to add, subtract, multiply and divide with the written methods, and know a square root as the operation that undoes squaring from square roots.
Does 802 − 47 come to 845?
Working 802 − 47 out again runs the same steps in the same order, so a slip in the exchanging is made again and the two attempts agree. Adding the 47 back to 845 tests the answer instead: a wrong answer carried backwards does not land on 802.
Example: 802 − 47 ≟ 845
Add back what was taken away.
- 845 + 47 The check starts from the answer, not from the numbers that made it.
- 845 + 47 = 892 Adding the 47 back repeats none of the original working.
- 892 ≠ 802 The subtraction started at 802, so 845 cannot be its answer.
The check does more than disagree: the gap is the size of the slip. 892 overshoots 802 by 90, so 845 is 90 above the true answer.
Ninety is nine tens and no ones, so the ones digit 5 is right and the slip is further left: 845 − 90 = 755, and 755 + 47 = 802 confirms it.
Which statement undoes the calculation
- 38 + 56 ≟ 94 94 − 56 = 38
- 94 − 56 ≟ 38 38 + 56 = 94
- 24 × 15 ≟ 360 360 ÷ 15 = 24
- 360 ÷ 15 ≟ 24 24 × 15 = 360
- 62 ≟ 36 36 = 6
Every one of those pairings is the same move: the two end numbers change places, and the operation turns into its inverse:
A finished subtraction
47 has been taken from 802, and 755 is the answer to test.
The inverse statement is not the only one that tests the answer: 94 − 56 = 38 is confirmed by 38 + 56 = 94 and equally by 94 − 38 = 56. 56 − 38 = 18 uses two of the same numbers and makes a fourth, so it tests nothing.
Example: 19 × 24 ≟ 456
Undo the multiplication with a division.
- 456 ÷ 24 Start from the answer and divide by one of the numbers.
- 456 ÷ 24 = 19 The other number comes back, so the multiplication holds.
When one inverse statement is not enough
47 ÷ 6 = 7 r 5 has two parts to its answer, the 7 whole groups and the 5 left over, and a check has to put both back.
Example: 47 ÷ 6 ≟ 7 r 5
Put the whole groups and the remainder back together.
- 7 × 6 = 42 The whole groups, multiplied back.
- 42 + 5 = 47 The remainder is added on, and 47 comes back.
What a root check leaves unsettled
A square root undoes a squaring only as far as size: 49 = 7 says nothing about whether 7 or −7 was squared:
Example: (−7)2 ≟ 49
Test the size with a root, then settle the sign separately.
- 49 = 7 Both 7² and (−7)² are 49, and the root sign names the positive one.
- 7 ≠ −7 Not a disagreement: −7 is the same size as 7, and size is all the root tests.
- The index 2 is even. An even index makes the answer positive whichever sign went in, so (−7)² = 49 is right.
A cube root keeps the sign of the number under it, so a cube needs no second step: −64 = −4 checks (−4)3 = −64 by itself. Only an even index gives the same result for a number and its negative; choosing between them is on positive and negative square roots.
Common mistakes
- Leaving the remainder out of the check For 38 ÷ 5 = 7 r 3, 7 × 5 = 35 multiplies the whole groups back and stops, so it lands on 35 and not on the 38 it started from; with the remainder back in, 7 × 5 + 3 = 38.
- Undoing with the wrong root A square root undoes a squaring, and 34 = 81 was a fourth power. 81 = 9 is 3 × 3, two of the four 3s in 81, so it does not return the 3 that was raised. 81 = 3 does.
- Working it out again instead of checking it Redoing 600 − 247 repeats the exchange slip, so 447 comes back a second time and looks confirmed; 447 + 247 = 694, which is not 600.
Summary
- Redoing a calculation repeats a slip in its method; undoing it does not.
- A check starts from the answer and applies the inverse operation.
- The gap between the check and the original number is the size of the slip.
- A remainder is part of the answer, so 7 × 6 + 5 = 47 and not 7 × 6 = 42.
- A square root gives back the size of a number and not its sign, so an even index leaves one more thing to settle; an odd index leaves nothing.