Why borrowing works

An explanation of why (not how) borrowing/regrouping works when subtracting numbers

Welcome to the presentation on
why, not how, borrowing works.

And I think this is very
important because a lot of

people who even know math
fairly well or have an advanced

degree still aren't completely
sure on why borrowing works.

That's the focus of
this presentation.

Let's say I have the
subtraction problem

1,000-- that's a 0.

1,005 minus 616.

What I'm going to do is I'm
going to write the same problem

in a slightly different way.

We could call this
the expanded form.

1,005-- what I'm going to do
is I'm going to separate

the digits out into
their respective places.

So that is equal to 1,000
plus let's say zero 100's

plus zero 10's plus 5.

1,005 is just 1,000
plus 0 plus 0 plus 5.

And then that's minus 616.

So that's minus 600
minus 10 minus 6.

616 could be rewritten
as 600 plus 10 plus 6.

And I put a minus there
because we're subtracting

the whole thing.

So let's do this problem.

Well, if you're familiar with
how you borrow is, this 5 is

less than this 6, so we have to
somehow make this 5 a bigger

number so that we could
subtract the 6 from it.

Well, we know from traditional
borrowing that we have to

borrow 1 from someplace and
make this it into a 15.

But what I want to see
actually, is understand where

that 1 or actually where
that 10 comes from.

Because if you're turning this
5 into a 15 you actually

have to add 10 to it.

Well, if we look at this top
number, the only place that

a 10 could come from is
here, is from this 1,000.

But what we're going to do
since this is the 1,000's

place, instead of borrowing 10
from here, which would make it

kind of a very messy problem,
I'm going to borrow

1,000 from here.

I'm going to get
rid of this 1,000.

And I have a 1,000 that
I took from this 1,000.

I have 1,000 now that
I can distribute into

these 3 buckets.

Into the 100's, 10's
and 1's buckets.

Well, we need 10 here,
so let's put 10 here.

So it's 10 plus 5
is equal to 15.

We got our 15.

If we took 10 from the 1,000
then we have 990 left.

So we could put 900
here and 90 here.

Notice, we just said-- so we
had 1,000 and we just rewrote

it as 900 plus 90 plus 10.

And we added this 10 to this 5.

And now we could do this
subtraction just how we

would do a normal problem.

15 minus 6 is 9.

90 minus 10 is 80.

900 minus 600 is 300.

So 300 plus 80 plus 9 is 389.

And let's see how we would have
done it traditionally and make

sure that it would have kind of
translated into the same way.

Well, the way I teach it and I
don't know if this is actually

the traditional way of teaching
borrowing, is I say, OK, I need

to turn this 5 into a 15.

So I have to borrow
a 1 from someplace.

Well, we know from this side of
the problem that we actually

borrowed a 10 because that's
why it turned to 15.

If we're going to borrow
1, I'd say, well, can I

borrow the 1 from the 0?

No.

Can I borrow the 1 from this 0?

No.

I could borrow it from
here, but I'm borrowing

it from 100, right?

So 100 minus 1 is 99.

So that's the how I do it.

And I say 15 minus 6 is 9.

9 minus 1 is 8.

And 9 minus 6 is 300.

So this way that I just did it
is clearly faster and, I guess

you could say it's easier, but
a lot of people might say, well

Sal, that looks like a
little bit of magic.

You just took that 5, put a 1
on it, and then you borrowed

a 1 from this 100 here.

But really, what I
did is right here.

I took 1,000 from this 1
and I redistributed that

1,000 amongst the 100's,
10's, and 1's place.

Let me do another example.

I think it might make it a
little bit more clearer

of why borrowing works.

Let me do a simpler problem.

I actually started off with a
problem that tends to confuse

the most number of people.

Let's say I had

732

minus-- Let
me do a fairly simple one.

Minus 23.

Sometimes those 3's
just come out weird.

Well, we just learned that's
the same thing as 700 plus

30 plus 2 minus 20 minus 3.

Well, we see this 2, 2 is less
than 3, so we can't subtract.

Wouldn't it be great if we
could get a 10 from someplace?

We could get a 10 from here.

We make this into 20 and add
the 10 to the 2 and we get 12.

And notice, 700 plus 20
plus 12 is still 732.

So we really didn't change
the number up top at all.

We just redistributed its
quantity amongst the

different places.

And now we're ready
to subtract.

12 minus 3 is 9.

20 minus 20 is 0 and then you
just bring down the 700.

You get 700 plus 0 plus 9,
which is the same thing as 709.

And that's the reason why
this borrowing will work.

Well, we say, oh, let's
borrow 1 from the 3.

Makes it a 2.

This becomes a 12.

And then we subtract.

9 0 7.

Let's do another
problem, one last one.

And once again, you don't
have to do it this way.

You don't have to every
time you do a subtraction

problem do it this way.

Although if you ever get
confused, you can do it this

way and you won't make a
mistake, and you'll actually

understand what you're doing.

But if you're on a test and you
have to do things really fast

you should do it the
conventional way.

But it takes a lot of practice
to make sure you never are

doing something improper.

And that's the problem.

People learn just the rules,
and then they forget the

rules, and then they
forgot how to do it.

If you learn what you're doing,
you'll never really forget it

because it should make
some sense to you.

Let's do another one.

If I had 512

minus 38

Well, let's keep doing it
that way I just showed you.

That's the same thing
as 500 plus 10 plus

2 minus 30 minus 8.

Well, 2 is less than 8.

I need a 10 from someplace.

Well, one option we can
do is we can just get

the 10 from here.

So then that becomes 0.

And then this will become a 12.

Notice that 500 plus 0 plus
12, same thing as 512 still.

So we could subtract.

12 minus 8 is 4.

But here we see this 0 is less
than 30, so we can't subtract.

But we can borrow from the 500.

Well, all we need is 100, so if
we turn this into 100 then we

took the 100 from the 500.

This becomes 400.

I just rewrote 500
as 400 plus 100.

Now I can subtract.

100 minus 30 is 70.

Bring down the 400.

And this is the
same thing as 474.

And the way you learn how to do
it in school is you say, oh,

well, 2 is less than 8,
so let me borrow the 1.

It becomes 12.

This becomes a 0.

0 is less than 3, so let
me borrow 1 from this 5.

Make this 4.

This becomes 10.

So then you say
12 minus 8 is 4.

10 minus 3 is 7 and
you bring down the 4.

Hopefully what I've done here
will give you an intuition

of why borrowing works.

And this is something that
actually I didn't quite

understand until a while after
I learned how to borrow.

And if you learned this, you'll
realize that what you're doing

here isn't really magic.

And hopefully, you'll never
forget what you're actually

doing and you can always
kind of think about what's

fundamentally happening to
the numbers when you borrow.

I hope you found that useful.

Talk to later.

Bye.