www.mathcentre.ac.uk/.../Rules%20of%20arithmetic.mp4

Today we're going to go over the
President Rules of arithmetic,

which allow us to workout
calculations involving brackets,

powers, division and
multiplication, addition and

Subtraction, and let us all
arrive at the same answer.

Then afterwards, we're going to
go on and do calculations

involving positive and negative

numbers. And use rules for
addition, subtraction,

multiplication and division.

But first, let us look at this

expression. It's 2 + 4 * 3 - 1
and let us work at night.

Well, we can work it out first
by moving from left to right,

and if we do that then we will
add, then multiply and then

subtract. If we do that, we get
2 + 4, which is 6 times by

three. Is it T minus one gives
us the answer 17.

But if we change the order, the
calculation and we use add

first, then subtract and then

multiply. What answer do we get?
Well, 2 + 4 is 6. Do they

subtract next 3 - 1 which is 2
so 6 * 2's answer is 12.

Or we could multiply first,
then add followed by subtract.

Well, if we multiply first,
that's 4 * 3, which is 12.

Do the add 2 + 12 which is 14.

Then subtract the answer is 13.

And so on.

And as you can see, we get
different answers according to

the order in which we do the
operations, so that's not very

good. So we need to have a
presidents in which we know the

order in which we do the

calculation. The order that most
people use is.

We do brackets first.

Followed by powers.

And then multiplication

Indovision. Then finally
addition and Subtraction.

And that's a lot to remember.

So there's an acronym to allow
you to remember it.

And that acronym
is barred maths.

BODMAS.
Be for brackets.

Oh for powers.

Date for

division. An multiplication
for the M.

And these two go together 'cause
they have the same priority

Avery Edition. And
As for subtraction.

And these two go together
because they have the same

priority. So if you remember Bob

Maths. FIFA brackets and
Overpowers followed by division

and multiplication, and then
finally addition and

Subtraction. If you want to
calculate any expression, use

that order and you won't go

wrong. So why don't we
look at some examples?

Take for example, the first one
that we started off with 2 + 4

times by 3 - 1. Now using board
maths we should have no problem

working night this expression we
do the Times first, so it's two

add 12 - 1 night. The addition
and subtraction of the same

order of priority. So we work
from left to right. That's 2 +

12 which is 14 - 1 which is 13.

Look at this next example.

Here we've got brackets and
using board mass brackets are

done first, so 3 + 5 is it,
so our expression becomes two

times it, which is 16. Can be

easier. What about the next one
9 - 6? But I've got to add the

plus one now in this case we've
got the subtract and the ad

together. So we've got the same
priority. The rule is work from

left to right, so we say 9 - 6
which is 3 add 1 which is 4.

The last two involving powers.

3 + 2 squared while using board
Mars again, we do the squares,

the powers first 2 squared is 4,
so our expression becomes 3 + 4

and 3 + 4 is 7.

And finally this expression.

3 + 2 all squared. Now we've got
powers, an brackets. But

remember using board Mads we do
what's inside the brackets

first, so it's 3 + 2 which is 5.

And then we have to square the
five which is 25.

Now in these two last examples
with the got the same numbers.

And we've got a square, but
you've got two totally

different answers. And the
reason why is in this one the

square is for the things
inside the bracket. The 3 + 2,

whereas in this one this
square relates to just the

two.

So remember when you're doing
any calculations which involve

operations such as brackets,
powers, division,

multiplication, subtraction and
addition, use board baths and

you won't go wrong.

What math means?

Brackets first, then powers
followed by multiplication and

division. And then addition to

Subtraction. And when you have
operations of the same priority.

He just work from left to right.

Nothing could be easier.

And now I will move on to
calculations involving positive

and negative numbers. Now what
are positive and negative

numbers? Well, if we take all
the real numbers except 0.

All real numbers.

Can be. Either
positive or

negative, and of
course except 0.

Now, where are those
numbers? Well, if we

look on a number line.

And position 0 all the positive
numbers are to the right.

And all the negative numbers are
to the left.

And we represent the numbers on
the number line like this

positive one, positive 2
positive 3 positive for.

Positive 5 and so on.

Negative one negative, two
negative, three negative,

four negative 5.

Notice how we've written in
numbers, the positive numbers

and the negative numbers. We
write positive three like this

and negative for like this. The
sign of the number is written as

a superscript. Now we do this.

To help our understanding so
that we don't confuse the sign

of the number with addition and

Subtraction. But of course with

practice. We drop this and we
just use the normal standard

notation. But for this
session I'm going to use the

superscripts just to help
our understanding.

But what about calculations
involving positive and negative

numbers? What about addition,
subtraction, multiplication and

division? Well, I would take
some examples. If I take these

two examples, negative four at

positive 5. And.

Positive for. Subtract

positive 9. If I have
those two calculations, what do

they work out to be well?

In the first instance, we can
use a number line to help us

evaluate them. And we have to
remember that using a number

line addition means you kind on
in that direction, and

subtraction means you kind back
in that direction.

So using those two little rules
will calculate this expression

and then this.

So take this one to begin with
negative 4. Add positive five.

Start at negative, 4 on the

number line. And kind on
five 12345, so it negative

4 add positive 5 gives
me the answer positive one.

For the other example, we've got
a subtraction, so we started

positive four, and we subtract

positive 9. To start a positive

for. And kind

back 91234.
56789 so

positive for

subtract. Positive 9 gives
you negative 5.

It could be simpler.

Now, to simplify matters a
little bit further, it really is

a fast to keep writing all these

signs. And because positive
numbers are the numbers that we

usually use in calculations, we
drop them. We dropped the sign.

So positive 5 can be written is
just five, and we know that

positive one is equal to 1 and
positive 9 has 9.

And it's understood by everyone
that where you have numbers with

no sign written there then they
are positive numbers.

And I'm going to use that.

In all my calculations following

on. But if you notice in
this calculation and in this

one. We've added and
subtracted positive numbers.

But how do we add and subtract

negative numbers? Well, try and
generate an easy route.

Going to look at some patterns
of both addition and Subtraction

and will start with addition.

And we'll start with something
that we know.

If we start with 5 + 2,
we know that is 7.

And then I'm going to write a
sequence of calculations which

follow on from this.

5 at one.

Is equal to 65. Add zero
is equal to 5.

Notice in this sequence of
addition the answers decreased

by one. As the numbers that we
add decreased by one.

So if we follow the pattern.

We take 5 add negative one
because negative one is 0.

Subtract 1 is one less than
zero, then it must equal for. If

we continue the pattern on.

And five add negative, two must
equal 3 and five ad.

Negative three must equal
two and five add negative.

Four must equal 1.

Not if we look at these

additions. These additions of
negative numbers we can actually

write these calculations as
subtractions of positive

numbers, so 5 add negative, one
would be the same as five

subtract 1 and you get the same

answer for. Similarly, 5 add
negative two can be written as

five subtract 2 and you get the
answer 35 add negative. Three

can be written as five, subtract

3. Give me the answer 2 and

then 5. Add negative four is
the same as five. Subtract

4, which is one.

So when we have the addition
of negative numbers.

That's the same as
the subtraction of

the positive number.

So I had these two examples.

It at negative 10.

And negative 9 at

negative 5. Using that

room. Hi, we calculate

the answer. Now we take it add
negative 10. That's the same as

it subtract 10.

I'm thinking about the number
line. He started it and you go

back 10. So you go back to 0
and then another two you get

to the answer negative 2.

And with this example.

Negative nine and negative five.
We can rewrite that as negative

9 subtract 5 again. Visualizing
it on the number line start at

negative 9. And you go
back five, go back five so your

land at negative 14.

So when we have the addition of
negative numbers, it's always

easier to change them into
subtraction of positive numbers.

But the one thing that we
haven't done is the subtraction

of negative numbers.

So again, I'd like to use a
pattern to help us workout the

rule that we're going to use.

If we start off with what we

know. 4 subtract 2 is 2.

So 4 subtract 1 is
3 and four subtract 0

is for.

Notice in this sequence of
Subtractions, the answers are

going up by one.

As the numbers that we're
subtracting decreased by one.

So if we continue the pattern in
the calculations, the next

calculation would be 4. Subtract
negative one and the answer will

be 5. You add one on to the
four to make 5, and four

subtract negative 2.

Will be 6 and four. Subtract
negative. Three would be 7.

And four subtract
negative four is it?

Now again. If we look
at these subtractions

of negative numbers.

What calculation would be easier
to do using the numbers but

still arriving at the same
answer when I think it's quite

obvious. For this one, it's for

AD one. And that gives us the
answer 5. The numbers of the

same, but the operation and
assign are different.

For this one it before add 2 and
that will give us 6.

For this one it before at three,
and that will give us 7.

And finally, for ad for which
will give us it.

So.

If we look at our pattern, we
can see that when you've got the

subtraction of a negative
number, that's the same as the

addition of a positive number.

So we'll use that room to
workout these calculations if we

take it, subtract negative 10.

And negative 6 subtract

negative 13. What answers do
we get if we use our room where

we change the subtraction to
addition of a positive number?

So is it plus 10? What could be
easier? That's 18.

This one you've got negative 6
add 13 slightly harder, but not

too difficult. They give
negative 6 and the number line

and go forward kind on 13. He
kind on 6 and then another

Seven. So the answer is positive
7 written like that.

So how can we combine all these
rooms together in a nice, easy

way for you to remember? Well,
this is one way.

If the operation in the sign are
the same like this.

Sam

The calculation works like
an addition.

Off a positive number.

If the operation and assign are
different like this.

Then the operation.

Works like this subtraction.

Off a positive number.

Now, if you remember those two
Golden rules, then the addition

and subtraction of positive and
negative numbers is dead easy.

Now we've talked about addition
and subtraction of positive and

negative numbers, but wanna bite

multiplication Indovision. Well,
we start with what we know. We

know how to multiply and divide
positive numbers. We know that

five times by 5 is 25.

An 5 / 5 is one

dead easy. But what about
the multiplication division

of negative numbers?

Well, I'll use patterns the way
I did before. In addition,

Subtraction. And I'll start off
with this calculation and

continue doing a sequence of
multiplications that involve

negative numbers, and we'll see
if we can get a rule coming out

from the pattern in the

calculations. For the next
calculation here, I'll say this

is 5 times by 4.

And we know that is 20.

Five times by 3:15,
five times two is

10. 5 * 1
is 5 and 5

* 0 is not.

I notice in these
multiplications the answers.

Are going down by 5 as the
number that we times by

decreases by one.

So the next calculation in the
sequence would be 5 times by

negative one. And if we use the
pattern that have just stated.

We subtract 5 from zero as we
subtract 5 from zero, we get

negative 5. The next calculation
would be 5 times by negative 2.

Negative 5 subtract
5 is negative 10.

Five times by negative 3.

Negative 10 subtract 5 negative

15. Will do just one more just
to see if we can spot the

pattern and the pattern is
working five times negative 4.

Take negative 15 and subtract 5.
That's negative 20.

So look at.

Our multiplications by a

negative number. We get
a negative number answer.

So when we multiply a negative
sorry, a negative by a positive.

Will always get a
negative answer.

And vice versa if we multiply a
negative by a positive.

Will get a negative.

So if I give you these two

examples. 6 times by
negative 5.

Steady, easy to workout. The
answer you just multiply the six

by the five to get 30. The signs
are different. Remember this is

positive 6 times by negative 5
the two signs are different, so

the answer is a negative number

negative 30. If we take
negative four times by three.

Multipliers normal 3/4 or 12.

And then take into account
the signs. This is a

negative. This is a
positive. 2 signs different.

So the answer is negative.

And the same goes for division.

And we can just double check
that by looking at these two

calculations. Negative 30
divided by positive six must

equal negative 5.

And negative 12 divided by
positive three must equal

negative 4.

So when the signs are different
and you multiply positive and

negative numbers together in
pairs, the answer will always be

a negative number.

But we haven't finished just
yet. What if you multiply and

divide by negative numbers?

What happens there?

Well, again, I'm going to use
patterns. I like using patterns

because it gives me a bit of
confidence that I'm doing things

correctly and I always start off
with things that I know. So if I

start off with negative five
times by positive 4.

I know the answer is negative 20
because the signs are different

and 5 * 4 is 20.

Next one in the sequence
negative five times by three.

That will give us negative 15
negative 5 * 2 negative, ten

negative 5 * 1 is negative,
five negative 5 * 0 is

equal to 0.

Now look at the pattern.

Is a bit like a dejavu. We've
done this before. Look at the

answers. You can see that we
are increasing by 5 each time

as we multiply by one less
each time. That seems a bit

odd, but stay with Maine.

The next multiplication, if we
keep the pattern in the

calculations, the same would be
negative five times by negative

one. And according to our
pattern, that will equal 5 more

than zero, which is positive 5.

Negative five times negative 2,
which is the next calculation in

sequence. That would be five
more than five, which is 10.

Negative five times by negative

3. Is 15th.

Negative five times negative
four is 20.

So. It's really great. See that
when we multiply negative

numbers together in pairs.

We get a positive answer.

Negative by negative gives it a
positive negative negative

positive, negative negative
positive negative negative

positive is not great. Dead
easy. So if I had this

calculation negative 6 times by
negative three, you multipliers

normal. 6 * 3 is it teen? And
the answer is positive positive

via team because it's two
negatives makes the positive.

And the same goes with
this one. If we had

negative 9 times by negative

229382. Two negatives.

The answer is a positive, so
again we get the same answer

it team but different numbers
in the calculation.

But remember where else did we
get a positive answer?

When you multiplied 2 positive

numbers together. So if you
multiplied 6 by three, you'll

get it team and when you
multiplied 9 by two you get it

too. So when the signs of

the same. Then the answer will
be positive whether they are

two negatives be multiplied
together or two positive

numbers being multiplied
together.

And that's a lot to take
in for multiplication and

division of negative numbers.

I'd like to summarize that bit
by again using a diagram when

the signs of the sea.

A positive times by a positive
or a negative times by a

negative. The

answer. Is

positive. Is a
positive number when the signs

are different. I positive
times by a negative or

a negative times by a

positive. The answer.

Is negative.

And the same goes the same
rules go if you divide.

So if
I had

these examples.
Negative 6 divided

by. Negative
2.

My answer will be do the
division first 6 / 2, which is 3

and then think about the signs
negative negative signs saying

so. The answer is positive.

And if we had this
division negative 12 divided by

positive 3. Do the division is
normal 12 / 3 is for.

Think of the signs
signs, different

negative and a
positive. So the

answer is negative.