Welcome to an introduction to decimal notation.
In this lesson we will read a number in decimal notation,
write a decimal using fractional notation, and also comparing order numbers
in decimal notation. Decimal notation is used to write numbers according to
place value in base ten. Decimal point is used to separate whole numbers
from numbers that are less than one. So looking at the place value chart below,
here's a decimal point. To the left we have the whole numbers.
We have the ones, tens, hundreds, thousands,
and ten thousands. To the right of the decimal point we have numbers that are
less than one. We have the tenths, hundredths,
thousandths, ten thousandths and so on place value.
To help us model decimals, we use the following base ten blocks.
We'll let this block be the unit block. And then because it takes ten flats to equal one
whole,
one flat is equal to one tenth as a fraction, or zero point one as a decimal.
It takes one hundred rods or one hundred longs to make one whole.
So one rod is equal to one hundredth as a fraction or zero point zero one as a decimal.
And it takes one thousand bits to make up one whole,
and therefore one bit is equal to one thousandth, as a fraction
or zero point zero, zero, one as a decimal.
So to the right of the decimal point we have the tenths, hundredths,
thousandths and ten thousandths place value.
Here we're asked to read the following numbers using place value,
and then write the decimal, using fraction notation.
So we first have four point three.
To help us better understand this let's represent four point three, using our base ten blocks,
as defined before. So because we have a four in the ones place value, we have four
ones.
And then because we have a three to the right of the decimal point, this represents
three tenths, which would be three flats.
Again, each of these flats is equal to one tenth, so we have
three tenths. This is actually how we say the number. We say four
and three tenths. Which means as a mixed number,
we would have four and three tenths.
If we're asked to express this
as a fraction though, we would express this as an improper fraction
because each one is equal to ten flats or ten tenths.
We have forty tenths, plus three tenths, or forty-three tenths.
Another way to find this improper fraction
is to multiply the denominator times a whole number.
Four times ten is forty. Add the numerator of three,
which gives is forty-three, giving us forty-three tenths.
Let's look at a second example. Here we have
two point one, two. So modeling this using our base ten blocks.
Here's our two. We have a
one in the tenths place value so we have one tenth or one flat.
And then we have a two in the hundreds place value.
So we have two longs or two rods, this is two
hundredths. So we have two and
one tenth and two hundredths. We normally don't read the number this way
looking at the values that are less than one, we obtain common units.
Which means we can perform an exchange for this one flat.
We can exchange this one flat for ten longs or ten rods,
which would look like this. So here we have two. Now, instead of having one flat we have
ten longs,
which shows us that one tenth is equal to
ten hundredths. And we still have two hundredths here.
Notice how we have a total of ten plus
two or twelve hundredths. This is how we say this number
two and twelve hundredths. Looking at the decimal
again, notice how we do have a twelve to the right of the decimal point,
and because the rightmost digit which is a two is in the hundreds place value.
Again we say two and twelve hundredths.
Which means is a mixed number we do have two and twelve hundredths.
But notice how this fraction does simplify.
Twelve and one hundred share a common factor of four.
So we could divide out the common factor of four.
So simplified, we'd have two,
and three twenty fifths. Now this is a mixed number so as a fraction,
our denominator would still be twenty-five, and the numerator would be
twenty-five times two, that's fifty, plus three that's fifty-three.
So two and twelve hundredths
as a fraction is fifty-three twenty fifths.
For our third example, we are given one point
one, two, five. So using our base ten blocks,
here's one. Because because we have a one in the tenths place value
we have one flat, here's the one ten.
Because we have a two in the hundredths place value,
we have two longs or two hundredths. And then we have a five in the
thousandths place value, so we have five bits.
So we have one
and one tenth, and two hundredths, and five thousandths,
but this is not how we normally say this number. If we take a look at all the values
that are less than one,
then we get common units. We want to convert all of these to
thousandths. So because it takes ten bits to equal
one long, each of these longs is ten
thousandths. So we have a total of twenty thousands,
meaning two hundredths is equal to twenty thousandths.
Because it
takes a hundred bits to equal one flat, one tenth is equal to
one hundred thousandths. So notice how we have one
and one hundred twenty-five thousandths.
So as a mixed number again we have one,
and one hundred twenty-five thousandths. Looking at our decimal again, notice how we do have the number one hundred twenty-five,
and because the rightmost digit is in the thousands place value
we read this, one and one hundred twenty-five
thousandths. Now this fraction does simplify.
One hundred twenty-five does divide evenly into one thousand.
So if we divide out the common factor of one hundred twenty-five,
it should be equal to
one an one eighth. Again eight times one hundred twenty-five is equal to one thousand.
So as a fraction, one and one eighth would have a denominator of
eight. The numerator would be eight times one,
which is eight plus one, which gives us
nine eights. Again one is equal to eight eighths.
Eight eighths plus one eighth, which gives us nine eighths
as the fraction for one and one hundred twenty-five
thousands. Okay, I hope you found this helpful.