8
$\begingroup$

We can multiply $a$ and $n$ by adding $a$ a total of $n$ times.

$ n \times a = a + a + a + \cdots +a$

Can we define division similarly using only addition or subtraction?

  • 0
    @Arkamis, what is this set whose supremum is $b^x$? Sounds very interesting.2014-12-25

4 Answers 4

23

To divide $60$ by $12$ using subtraction:

$\begin{align*} &60-12=48\qquad\text{count }1\\ &48-12=36\qquad\text{count }2\\ &36-12=24\qquad\text{count }3\\ &24-12=12\qquad\text{count }4\\ &12-12=0\qquad\;\text{ count }5\;. \end{align*}$

Thus, $60\div 12=5$.

You can even handle remainders:

$\begin{align*} &64-12=52\qquad\text{count }1\\ &52-12=40\qquad\text{count }2\\ &40-12=28\qquad\text{count }3\\ &28-12=16\qquad\text{count }4\\ &16-12=4\qquad\;\text{ count }5\;. \end{align*}$

$4<12$, so $64\div 12$ is $5$ with a remainder of $4$.

  • 0
    Seems a bit arbitrary that we always want to stop at zero. Nothing more than definition right?2016-06-03
10

If $n$ is divisible by $b$ ($\frac{n}{b}$ is a whole number), then keep doing $n - b - b - b - b - b - \cdots - b$ until the value of that is $0$. The number of times you subtract $b$ is the answer. For example, $\frac{20}{4} \rightarrow 20 - 4 - 4 - 4 - 4 - 4$. We subtracted '$4$' five times, so the answer is $5$.

4

You can also use additions. One should use results from intermediate calculations to speed up.

Let us divide 63 by 12. $ \begin{split} 12+12=24,&\qquad\textrm{count }1+1=2\\ 24+24=48,&\qquad\textrm{count }2+2=4\\ 48+24=72,&\qquad\textrm{count }4+2=6\textrm{ (exceeded 63)}\\ 48+12=60,&\qquad\textrm{count }4+1=5\textrm{ (so we try adding less)}\\ 63-60=3,&\qquad\textrm{(calculation of the remainder)}\\ \end{split} $

0

You can define division as repeated subtraction:${72\over 9}=72-9-9-9-9-9-9-9-9$Subtracting by $9$ eight times is the same as subtracting by $72$ since $9\cdot8=72$. So, the answer is $8$. Also, this is why ${n\over a}=n-a-a-a-a\cdots$ for whatever whole number $a$ is other than zero.

If you have a remainder, then you just do this:${13\over 2}=13-2-2-2-2-2-2-1$as you just saw, subtracting by $2$ six times is the same as subtracting by $12$ since $2\cdot6=12$, but there's a remainder of $1$ being sutracted, so it's the same as subtracting by $13$ since $2\cdot6+1=13$, so the answer is $6$ R$1$ or $6.5$.

  • 0
    Okay, I edited it so it would make more sense.2015-01-24