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.