Can anyone explain to me "subtraction by addition" in a visual way?

The steps say:

1. Take the "complement" of the number we are subtracting
2. Add it to to the number we are subtracting from
3. Discard the extra "1" on the left

Example: $9 - 7$

1. $7$'s complement is $3$
2. Therefore $9 + 3 = 12$
3. We discard the $1$ on the left and the answer is $2$

• What is meant by complement here? Is there a particular step (either in the example or the explanation) that doesn't make sense? – Joe Tait Jun 26 '14 at 13:02
• That's nothing compared to "addition by subtraction": To calculate $9+7$, you subtract $100-9=91$, then you subtract $91-7=84$, and then you subtract again, $100-84=16$. Useless, but somewhat funny. – Joker_vD Jun 26 '14 at 14:39
• @Joker_vD I still don't know what is so wrong with the traditional carry/borrow method that motivates educators to constantly come up with convoluted and crippled "algorithms" to perform such a simple task. – Thomas Jun 26 '14 at 14:42
• @Thomas If nothing else, it does provide an interesting analogue to two's complement arithmetic on binary computers. Perhaps electronics students could find some insight from this. – David Zhang Jun 26 '14 at 22:26
• For this to work it is important that you stay within a given number of digits and discard those outside. Otherwise 9+3 will be 12 and not 2. – Thorbjørn Ravn Andersen Jun 27 '14 at 12:56

As you have asked for a "visual" explanation: The "complement" of $7$ is the number of steps it needs to reach $10$ (that is $3$). But on the way to reach $10$, it reaches $9$ first. Thus, of the $3$ steps, $2$ steps are needed to reach $9$, and the remaining $1$ step takes $9$ to $10$. So, when you add $3$ to $9$, the first thing that happens is that $9$ itself reaches $10$ in that $1$ step, and then it goes above $10$ with the remaining $2$ steps. When you remove $10$ from this answer (by discarding the "extra $1$ on the left"), you get the $2$ steps which $7$ actually takes to reach $9$.

It was amusing to explain this, but I agree with Thomas in finding this method unnecessary and crippled.

• I think some explanation for where 10 comes from is warranted. Just the smallest multiple of 10 greater than the first operand? – jpmc26 Jun 27 '14 at 2:42
• @jpmc26 Yes, because the last step given by the OP is "discard the extra $1$". So the "complement" is defined as the difference between a multiple of $10$ and the operand. – M. Vinay Jun 27 '14 at 2:46
• The fact I even thought that was a legitimate question shows how confusing this method is if you don't take the sane, mathematical approach like @Fabien's answer. =) – jpmc26 Jun 27 '14 at 2:48
• @jpmc26 Indeed, that's short and simple. I don't know why the OP wants a "visual" explanation, because there's actually nothing special about the number $10$ considered as a number (when you see $10$ objects, for example). The only reason $10$ is used in this method is because of the way we represent numbers in the decimal system. The method would use complement with respect to $11$, if we were using base $11$. But when numbers are represented "visually" as numbers of objects, there is no base involved. – M. Vinay Jun 27 '14 at 2:57

$$9-7=9-(10-3)=9+3-10=12-10=2$$

• wow thanks. now i understand =) – afgphoenix Jun 26 '14 at 12:44
• @afgphoenix, my pleasure ;) – Fabien Jun 26 '14 at 12:46
• @afgphoenix if the answer is good enough, you can accept it by clicking on a tickmark to the left of it. – Ruslan Jun 26 '14 at 13:52

$$9-7 = 10-8 = 11-9 = 12 - 10$$

The method you are using is what we call Complement's Method or Method of Complement.

This method is implemented most commonly in digital computer to perform binary arithmetic.

## Processes in complements method (of decimal number)

• negative number is represented in complemented form of either 9 or 10.
For example,if a number is -15,
9's complement of 15 is 84, because 9-1=8 and 9-5=4
10's complement of 15 is 84 + 1 i.e. 85.

Note: The nines' complement of a decimal digit is the number that must be added to it to produce 9; the complement of 3 is 6, the complement of 7 is 2, and so on. 10's complement is 9's complement plus 1.

• positive number is left as it is
• addition is performed between the positive number and complemented form of negative number
• if 9's complement is performed, then 1 is added to final answer
if 10's complement is performed, the sum is left as it is.
• if extra digit appears in the final answer, it is omitted.

For example, for 50 - 5

9's complement:

• 50 - 5 → 50 + 94 (5 is taken as 05 as highest number of digit is 2)
• 50 + 94 = 144
• 144 + 1 = 145
• 145 → 45

10's complement:

• 50 - 5 → 50 + 94 + 1 → 50 + 95
• 50 + 95 = 145
• 145 → 45

I've been learning how to use the Japanese abacus for some years. So I'm used to calculate with complements all the time both when I add numbers or when I subsctract.

But this example "9 - 7" doesn't look particularly good to me. Because if I've got 9 beads on a row I don't need any trick to remove 7 beads: I just put my fingers on them and when I remove 7 beads, I've left with 2, end of story.

But if the exemple were "10 -7" and I have 1 bead on the tens and no beads on the one's, then I can't remove 7 beads from the one's row. So here it makes totally sense using complements.

I will borrow one from the tens row (by removing it) and then add 3 on the ones row, because 3 is the complement of 7 .

if I have the number 10 and I remove the one on the tenths row and I add 3 on the units row, the final result is = 3