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.