It’s Professor Dave, let’s talk about decimals.
We just learned about place values, so we know about the units place, tens place, hundreds
place, and so on, ad infinitum.
But what about all the fractions we learned about?
Going back to the pizzas, say they are cut into ten slices.
What if we have one and one tenth pizzas, is there a way to represent that without having
to use an improper fraction or mixed number?
There certainly is, all we have to do is extend the place values we know to represent these
fractions of one as well.
This is achieved with the decimal point.
Instead of calling one slice one tenth, we can call it zero point one, since this is
the tenths place.
Two slices is zero point two, all the way up to ten slices, which would give us one
whole pizza.
In this way, if we want to represent a number that is somewhere in between two whole numbers,
like the way one and one tenth pizzas is more than one pizza but less than two, we simply
place a dot to the right of the units place, and then include as many place values as we
need to represent the number.
In this case, we just need the tenths place; so 1.1 will suffice, since we have the one
in the units place, and a one in the tenths place, indicating that there is one tenth,
or one slice, out of the second whole number.
The place values to the right of the decimal work just like the regular ones, in that as
we go to the left each place value is ten times greater than the previous, while as
we go to the right, each place value is ten times smaller.
So after the tenths place, we get the hundredths place, thousandths place, and so forth.
That means this approach would work even if the pizza were cut up into a hundred slices,
as each slice, or perhaps more easily visualized, each of these one hundred tiny squares, would
be zero point zero one.
If the large square were cut up into a thousand squares, each square would be zero point zero
zero one, with the one in the thousandths place.
So we are beginning to see how place values report the magnitude of something, The number
one is the same symbol no matter how we use it, but it has the capacity to report enormous
values as well as values that represent tiny fractions of one, all depending on the place
value it inhabits.
We can do operations with decimal values just as we would normal numbers; we just have to
line up the place values.
2.3 plus 1.4 involves adding the tenths together to get seven tenths, and adding the units
together to get three units, so 3.7.
This is the same way we would add twenty-three and fourteen together; we just have a decimal
place present which alters the magnitude of the numbers.
Later we will learn all about how to convert between fractions and decimals, but for now,
let’s check comprehension.