www.mathcentre.ac.uk/.../Decimals.mp4

In this tutorial, we're going to
look at the meaning of decimals

and their link to fractions.

Then we'll have a look at
rounding to decimal places

an to significant figures,
and then we'll take take a

look at irrational numbers.

So decimals, what does it mean?
Well, the word decimal means

connected with 10.

And we use the decimal number

system. To write all our numbers
from the very smallest up to the

very largest. Well, let's have a
look at how that works.

We use the
digits 012.

34
5678 and

9.

And those 10 digits are the only
ones that we use in our number

system. A decimal number system.

So let's take a number,
for example 12.

The two represents 2 units.

But I want in this case.

Doesn't represent one unit.

But it stands for a 10. So
what we've got here is a two.

The units plus the one
representing a 10.

So the digits zero to 9 are put
in different places to form our

numbers in our number system.

Let's have a look at how our
number system is constructed.

Let's start with our units.

As I units and.

As we move.

In this direction.

This column represents are 10s.

And we've got 10 times bigger. 1
* 10 gives us the 10.

As we move again.

We multiplied by 10 again.

And we come to our
hundreds column.

If we continue in the same way,
multiplying by 10.

We get to our thousands.

And again. Will now be one
followed by 4 zeros.

Our 10,000.

Let's do a couple more.

Multiply by 10 again.

And this time we
have 100,000.

And the last one will
look at for now.

Multiplied by 10 again is our
one followed by 6 zeros.

Oh, just about squeeze them in,
which is our million.

Now let's have a look.
What happens when we

get smaller than one?

Well.

What happened?

Going in this direction.

Well, if we want to go from a

million. Like 200,000 with
dividing by 10.

So if we go this way, we're
doing the reverse process

instead of multiplying by 10
with dividing by 10.

Let's do that all the way down.

210 / 10.

Gives us out one.

So if we divide 1 by 10.

I'm going to write it as a
fraction. We get 110th.

Well, let's divide by 10 again.

On this time we
get one hundreds.

On dividing by 10 again.

We get one thousands.

And if we just complete
what would happen here if

we started with one
thousandth and multiplied

by 10, we would get out
one hundreds and one

hundreds multiplied by 10
gets us to a 10th and

110th multiplied by 10,
gets us to one.

OK, let's have a look at
putting some numbers in

our place value system.

Just.

Put the chart on there.

OK, let's have a look at the
number 27 #27 is going to be.

Two in the 10s column and Seven
in the units.

27

531 So
we've got five hundreds.

Three 10s.

And one unit.

What about 50?

Well, 50 is five 10s.

But we can't just leave it as a

5 be'cause. That could
represent. As it stands now, 5

units we need to signify we need
to hold the place, the place

value of this five because it's
in the 10s column. So what we

need to do is to put a zero in
the units column.

So our 50.

What about 6000?

Let's put six in the thousands.

And again we have no digits
in our hundreds or 10s or I

units, but we need to put
zeros in there again to

hold the place value for
this six so that we know it

represents six thousands.

What about 207?

Well, 200 goes in the hundreds

column. Seven goes
in the units column.

There were no 10s.

So again, to hold the place
value to make sure this two is

in the hundreds column, we need
to show that there's no 10s, so

207, two, 07.

Let's look at another
one. Larger #120 Seven

1395. So the 100
Thousands 127 thousand. So

that's two in the
10 thousands, Seven in

the thousands and 395.

Let's move on to a
decimal number, let's say 6.392.

Well, six stands for six
units, but what do I do

with the .392?

Well up here you can
see that I've written.

The numbers that are smaller
than a whole one as fractions.

And in the decimal number

system. We've got a decimal
point and our not to 9 digits

again, so I need to put my
decimal point in.

And that goes in there between
the units and the tents, showing

that anything that comes
afterwards is a part of a whole.

So I'll put my point in there.

And the number I had was
6.392. So the three is 3/10,

the nine is 9 hundreds.

And the two is 2000s.

So the point shows us that the
part coming afterwards.

Is less than whole one
part of a whole 1?

Let's have a look at two more on

our chart. What about
.5?

Now .5.

We could put the point in and
put out five in its 5/10 a half.

Now. .5 I've said
point 5.5 is Acceptible.

But it's really useful,
especially when we're writing it

for clarity. If we put that zero
in the units column, it's not

needed strictly to hold the
place value, because we've got

the decimal point.

But to save it, getting the
decimal point, getting lost or

not seen or seen as a smudge on
the paper when it's written,

then it's very useful to put
that zero in the unit.

This column OK, so I'd
advise writing at 0.5

instead of just .5.

One more decimal

12.027. So 12 outside, one
in our 10s.

Too, and I units a decimal point

again. And 027 so that means
we've got none in the tents, two

in the hundreds and 7th in the

thousands. Now we're going to go
on to have a look at some

calculations, so I just remove
this part of the chart.

And put another
one in its place.

Now let's look at.

34 *
10.

Well, let's put 34 on our chart.

And we're going to
multiply it by 10.

So our four is going to get 10
times bigger, would be

multiplied by 10 and it's going
to move to the 10s column.

R3 is in the 10s column at the
moment. It's going to be made 10

times bigger so it's going to
become three hundreds.

Now to keep our place value.

We need to put a
zero in the units.

That's actually come from here
because we've got zero tenths.

We don't bother writing that
with the number 34. We don't

need to write it as 34.0, but
there is a zero here. And

obviously, when it becomes 10
times bigger, we need to 0 here,

but we must hold the place
value. So 34 * 10 gives us 340.

Let's have a look at
another one, this time 0.507.

And this time.

Just write it here.

Let's multiply by 100.

Well, let's have a look at this

zero. We multiply it by 10. It
moves to the tents column. If we

multiply it by 10 again.

It would move to the
hundreds column.

I just write it in for
the moment.

I5 here it's in the tents
column. We're going to

multiply it by 100, so we
multiply it by 10 and then

another 10. So that's going
to move to the 10s column.

The zero here is going to be
multiplied by 100, so

multiplied by 10 it moves
here and 10 again. It moves

to the units column.

And our Seven in the
thousands column multiplied

by 10, it moves to the
hundreds multiplied by 10.

Again, it moves to the tents.

Put a decimal point in.

So what we've got is
an answer of 50.7.

And you can see here this is the
case where we didn't really need

that zero, but I wrote it in for
clarity, but when we multiply by

100, we certainly don't need it.
'cause we don't say numbers as

no hundreds and 50.7.

So we can leave that out.

Let's have a look at dividing.
Now let's try 127.

.5. Divided by
10.

Let's put 127.

.5. Now we're going to
divide by 10, so we're

going to move the other way
in our system.

So. Can start from this and all
this and it doesn't matter. I

think I'll start from this end.

So our 5/10 with dividing by
10, so it's going to move to

the 500th's Column.

Are 7 Seven units divided by
10 becomes 7/10 and decimal

points always in the same place?

Between our units in our tents.

The two represents 10s,
so let's divide by 10

and it becomes 2 units.

Hundreds we have 100 / 10 and it
becomes a 10.

So 127.5 /
10 is 12.75.

Now I put on a new sheet of
paper to do some more

examples and you'll also see
that I've added one more

column heading here, one over
10,110 thousandth, and that's

because I'm going to need it
for the next example.

So let's have a
look at 2.3 /

1000.

So let's put 2.3 on our chart
and we're going to divide it by

1000. Well, the
1000 is the same.

As 10 * 10 * 10.

To make sure it's in a bracket
here, because we're dividing by

the whole 1000 so we have to do
the multiplication here first

before we do this division.

And that's actually
the same as 2.3 /

10 to the power 3.

So we have a link here without

indices. So let's actually do

that division now. Well, if we
take out 3/10 / 10, it moves

to the hundreds column divided
by 10. Again it goes to the

thousands column and divide by
10. Again it moves to the 10

thousandth's Column.

And there are two in the
units column divide by 10 and

again and again and it moves
to the thousands column.

A decimal point always
stays in the same place.

Now to hold the place value of
these two digits, we need to put

in our zeros. We've got no
hundreds and no tents.

Again, it's not essential that
we put out zero in our units,

but it helps for the clarity.

So I answer
is 0.0023.

Now let's have a look at just
two more examples, where we've

used powers of 10.

Let's have a look at 7.1 * 10
to the power 5.

Well, here's our 7.1.

Times 10 to the power 5 means
that are Seven is going to be

multiplied by 10:00 and 10:00
and 10:00 at 3:00 and 10:00 and

10:00. That's five times. So are
Seven is going to end up in the

100 Thousands Column.

Another one for multiple .1.
Sorry at 110th is going to

be multiplied by 10 five times
12345 so it ends up in

the 10 Thousands Column.

Now again, everything works from
our decimal point. The

changeover between our units in
our tents, so we must hold the

place value of this Seven and
this one. So that means we need

to fill in with zeros in all
these other columns.

So we have an
answer of 710,000.

And let's have a look 7.1 again,
let's use that.

But this time we're going to
multiply it by 10 to the power

of minus three.

Not power of minus 3
means we're Dividing.

So we're going to divide by 10.

Divide by 10. / 10 three times.

So 110th is going to
move 123 places in this

direction. And our Seven will
move 123 places in the same

direction. So they end up in
the thousands and the 10

thousandths column. Again,
decimal point there, and we

need to hold the place value.
So we need to put zeros in

these two columns and again
for clarity, a zero in our

units column. So 7.1 * 10 to
the minus three is 0.0071.

OK, let's have a look.

Now it out link with fractions,
I just need to remove these and

then we can turn the page.

Let's take 0.2.

Well I .2 is in
our tents column.

So that's exactly
the same as 2/10.

Now let's just go stage further
with our fraction and put it in

its lowest form. So that's
exactly the same as one 5th,

so .2 is 2/10, which is 1/5.
So from our place value chart

we can go directly to
fractions because we know that

this column represents tents.

Let's have a
look at 0.25.

Well, our two again is in the
10th column. That's 2/10.

But what we have In addition.

Is 5. In the hundreds column.

Well, if we had two tents and
five hundreds, we put them over

a common denominator of 100 and
we have 25 hundredths.

And if we put that in its
lowest form, 25 goes into 104

times. So that's exactly the
same as a quarter.

Let's try another one. Let's
take a bigger #134.

.526

So how do we turn that into a
fraction? Well 134.

Is a whole number.

And here the .5
is 5/10. So we've

got 134 + 5/10.

+2 hundreds.

Plus on the six is in
the thousands column, so

it's plus six thousands.

Now if we put these over a
common denominator, we got 134.

And our common denominator here
will be 1000.

So we've actually got
526 thousandths.

And again, if we put that in its
lowest form that actually both.

Divide by two.

So we get 263.

Over 500

that was changing a decimal into

a fraction. How about going the
other way? How about turning a

fraction into a decimal?

Well, let's have a look at
some examples.

Half.

This means we've got one whole 1
divided up into two pieces. So

we do one.

Divided by two.

We get 0.5.

We just carry out the division.

3/4

3 / 4

and that gives us 0.75.

So we look at one
third. We do 1 / 3.

And what we find is that we get
0.333333 and so on. It goes on

forever. Now. This is where you
need to take care when you're

using a Calculator that you spot
the recurring decimals that

they're going on forever and you
need to write them either then

to a certain degree of accuracy,
or we have a way of writing.

Recurring decimals 0.3 in this
case with a dot centrali over

the three. And that means
0.3333. Two with the three

going on forever.

Let's look now what happens
if we have a mixed fraction.

So let's say we have two and
five sixths.

Well.

That's like saying
we've got 2 + 5 six.

So what we're actually going to
do is the 5 divided by the six

first. And then we'll add the
two whole ones on afterwards.

5 / 6

is actually 0.83333333
and the three

keeps reccuring. So we
can write it with our

dot over the three.

Plus the two.

So two and five 6 written as
a decimal is 2.8, three with

three recurring.

OK, that leads us nicely into
doing some rounding so that we

can express this sort of number
and any other number for that

matter, to a certain degree of
accuracy, so we don't have to

write lots of decimal places.

Now there's two ways we can do
it. We can round to the number

of decimal places, or we can
round to a number of what we

call significant figures.

Let's look at decimal
places first of all.

Let's take an example.

32
.7914.

And let's say I want
the answer written.

Right to one decimal place
and often decimal places

abbreviated to DP.

Now the wait to do this one
decimal place, so we just want

to really chop off.

The rest of it, but we have to
do it with a bit of care. So

what I'm going to do is put a

line down. After the
1st decimal place.

Now this is where we have to be
careful, because depending on

what digit comes after the
Seven, as to whether we need to

change it or not.

Now let's have a look at the
rules for whether we change it.

If.

This digit is
a 0123 or

4.

There will be no
change to the 7.

But if it's
a 5678 or

9. Then we need to
change that 7.

Well, let's have a look. We've

got a 9. So that's Seven is
going to be affected.

Now, why is it affected?
Well, if we look at

this, 79 is actually
closer to 32.8 then it

is to 32.7.

And that's why we've got these
digits. The 5678 or 9.

Having an effect on here
be'cause, it's actually closer

to the next digit up in this

position. If it had been a
0123 or 4.

Then it would have been closer
to this number being 32.7 then

to 32.8. So that's why we're
doing it. So in this case to

one decimal place, the
number is 32.8.

Let's have a look at some
more examples now.

Let's write 15.

.2172.

And we're going to write it.

2 three decimal places.

Well, again, I'm going to take
my line. 3 decimal places, 123

places and put the line where I
want to chop it off.

And this is where I need to look

at the digit. After the line and
see if it's going to affect this

digit before the line. In this
case it's a 2.

2 means there's no change.

So 15.2172.
Written to three decimal

places is 15.217.

Couple more examples.
Let's have a look

at 0.315.

And this one were going to write

to. 2 decimal places.

So I need to put my line 1 two
decimal places. It's between the

2nd and the 3rd.

And this is the digit I need to
look at to see if it's going to

affect this one here. That comes
before the line. Well, it's a 5.

And here we have a five that
says it's to be changed.

So to two decimal places.

This number will be
written as 0.32.

Well, you might say
that actually 0.315 is

exactly between 0.31 and

0.32. So why should it change?
Why should it go up? Well, it is

exactly in the middle. If there
are any other digits here,

obviously it would be closer to
0.32, so it would go up.

But we have a mathematical
convention and what we say is

that if it's a five, it goes up.

Because if there are any other
digits, it will go up anyway.

So we won't make an exception
for when there aren't any other

digits will stick to one rule if
it's a 5 or 6789, it affects

this one. So to two
decimal places, 0.315 is 0.32.

OK, one more example which
will actually write to three

different decimal places. Let's
look at 6.2549.

We're going to write it to one

decimal place. Then we'll write
it to two decimal places.

And then will write it
to 3 decimal places.

So to one decimal place.

One place we put our line. This
is the digit were considering.

It's a 5, so that means it will
affect this two. So to one

decimal place. The
answer is 6.3.

To two decimal places? Well,
let's write the number again.

To two decimal places, 12 align
goes down. This is the digit we

need to consider. This time it's
a four. That means it's going to

have no effect on the digit
before the line, so our answer

is 6.25 to 2 decimal places.

For the three decimal
places, let's write

our number out again.

This time our line goes here.

And this is the digit we need to
consider to see if it's going to

have any effect on this digit.

It's a 9 so it is for
9 is going to be closer to

this being a 5. So answer to
3 decimal places 6.255.

Now let's have a look
at significant figures.

Now the rounding process is
exactly the same as we've been

doing, but it's where you start
that's different. Instead of

starting at the decimal point in
counting decimal places, we're

going to start at the leftmost
non 0 digit.

Let's look at some examples.

27.3721

and we want to write it too.

Three significant figures, and
we abbreviate significant

figures to SF.

Same process as before, so I
need my line, but this time I'm

not starting here at the decimal
point. I'm starting at the

leftmost digit, so that's the
two that is not zero. Still the

two and I count 3 significant
figures 123. So here's my line.

Again, the next digit from my
line is the one I need to

consider. Is it going to have an
effect on this digit?

Well, in this case it's a 7, so
yes, it is. It's going to change

that to four because it's
actually closer to it being a

four. So to three
significant figures, my

answer is 27.4.

Let's do some more examples.

This time
I have

0.005214.

And I want to write it to
just one significant figure.

Right, one significant figure
that's one digit. I go to the

left hand end.

But it must be the first non 0

digit. So this is not a
significant figure, neither is

this. Neither is this. Here's my
first non 0 so it's the left

hand most non 0.

Digit, so there's my line.
I just want one

significant figure.

So it's this digit we look at to
see if it's going to affect the

digit. The other side of the

line. It's a two, so it's
not going to have any

effect on this digit.

So answer writing this number
to one significant figure is

0.00. 5.

Let's look at one more example.
This time a big number.

27 million.

400 and
13,200.

I'm going to write this one
to two significant figures.

So where am I going to put my
line? I go to the left and then

the largest digit that's not
zero and I count to 1. Two,

there's my line, so this is the
digit I need to look at to see

if it has any effect on this
digit. It's a full, so that

means it's not going to have an

effect. So to two significant
figures, I want the two in the

7. But to hold the place value
of the fact that this is 27

million. I need to put zeros in
all the other columns.

So 27 million, 400 and 13,000
and 200 to two significant

figures. Is 27 million.

Now let's have a look
at a rational numbers.

Well, what are they? Well,
let's write out the word

first of all, irrational.

Numbers.

Well, irrational numbers are
numbers that cannot be expressed

as a fraction a divided by
B where A&B.

Are integers.

Get my red pen 'cause they
cannot be expressed. There can't

be written in that way.

Well, what does that mean?
Well, let's have a look at

some examples.

An example of an irrational
number is pie.

Now, pie, written as a decimal.

Is 3.1415926 and
it goes on

forever. It doesn't
repeat in any

pattern.

And so it can't be expressed as

a fraction. It's an
irrational number.

Another example is the square
root of 2.

If you put two in your
Calculator, press the

square root button.

You'll get

1.4142. 1, three

and again. The numbers will
continue that go on forever.

They won't repeat.

So again, it can't be
expressed as a fraction.

It's an irrational number.

Other examples? The
square root of 3.

Can this one is
1.73205 and again

continues on forever with
no repeating pattern?

And one more example is E.

And that's 2.718281. And
then again, continues. Going

on forever. It doesn't
terminate, and there's no

repeating pattern.

Now it's worth just mentioning.

Recurring decimals, because
sometimes people think that some

of that recurring decimals are
irrational numbers there, not

because for example, 0.3
recurring goes on forever that

we write a 0.3 without dot.

In fraction form is 1/3.

So it very much can, and in fact
more accurately be written as a

fraction, a overbee where A&B

are integers. So recurring
decimals are rational

numbers, and it's these sorts
of numbers where the digits

don't recur in any pattern
and they go on forever that

are irrational numbers.