How does long division work?

Question

Can someone give me a real life example of division in practice?? How does that work?

In: Mathematics

Answer 1

It takes advantage on how we write numbers.

Lets do 123/5 with the same steps as in long division but in more rigorous fashion.

123 can be written as 1×100 + 2×10 + 3×1 (or with exponents as 1×10^2 + 2×10^1 + 3×10^(0). We’ll need this later).

So now we have ( 1×100 + 2×10 + 3×1 ) / 5.

We can write this as

( 1×100 ) / 5 + ( 2×10 ) / 5 + ( 3×1 ) / 5

And further.

1/5 × 100 + 2/5 × 10 + 3/5 × 1

Now we can start doing the divisions.

1/5 would result in some fraction. We do not accept any fractions here. So lets go back few steps and combine the 100 term and 10 terms together.

( 1×100 + 2×10 + 3×1 ) / 5 = ( 12×10 + 3×1 ) / 5. = 12/5 × 10 + 3/5 × 1

12/5 is 2 and leftover 2/5. Lets keep the 2 here and move the leftover to the next term. So 12/5 ×10 = 2×10 + 2/5×10.

12/5 × 10 + 3/5 × 1 = 2×10 + 2/5 × 10 + 3/5 ×1 = 2×10 + 23/5 × 1

23/5 is 4 and 3/5 as leftover.

2×10 + 23/5 × 1 = 2×10 + 4×1 + 3/5 × 1

Again lets keep the 4 here and move 3/5 to the next term.
But we do not have any terms left? We can just add more.

At the beginnign we wrote the number using decreasing powers of 10 (10^(2), 10^(1), 10^(0)). So we can just add 0×10^-1 (=0×0.1) in there! (it is just zero, we can add as many zeros to number as we like)

2×10 + 4×1 + 3/5 × 1 + 0×10^-1 = 2×10 + 4×1 + 30/5 ×10^-1

30/5 is nice even 6.

So our result is

2×10 + 4×1 + 6 ×10^-1

Now we just turn this back into normal number.

24.6

Long division does these steps. (lets see how long division works with reddit formatting…)

5|123
-0 5 doesn’t go into 1.
12 carry the 1 to the next term.
-10 5 goes into 12 2 times. 2×10 goes into the result so remove 2×5 from here.
23 And carry the remaining 2 to next term.
-20 5 goes into 23 4 times. 4×1 goes to the result so remove the 4×5 from here.
30 And carry the remaining 3 to the next term.
-30 5 goes into 30 6 times. 6×0.1 goes to the result so remove the 6×5 from here.
00 No more remainders to carry so we are done.
Result is 2×10 + 4×1 + 6×0.1 = 24.6