Okay… in the last section, we learned one of the most important things about fractions!
We learned that they are numbers that are written like division problems.
We also know that fractions can be used to represent smaller parts of things.
Now since a fraction is just a division problem, if we wanted to,
we could go ahead and do the division and get an answer.
And that answer would be a regular number that‘s the value of the fraction.
Let’s look at this fraction for example: ten-fifths
That’s the same as the division problem 10 divided by 5.
Now we know that 10 divided by 5 is 2, so the value of this fraction is 2.
So what’s the deal? The value ‘2’ doesn’t seem like a smaller part of something!
In fact… it seems like TWO of something!
Aren’t fractions supposed to have a value smaller than one?
Here’s the deal…
If the top number of a fraction is bigger than the bottom number,
then the value of the fraction will be greater than one.
But, if the top number is smaller than the bottom number,
then the value of the fraction will be smaller than one.
But if we divide a smaller number by a bigger number and get a value that’s smaller than one,
then how are we going to write that value down like a regular number?
Isn’t ‘1’ the smallest number we can write? (well… besides zero)
Luckily the answer is no.
You see, we can write values that are smaller than one by using something called ‘decimals numbers’.
To understand how decimal numbers work, let’s first review how we write down regular numbers that don’t involve fractions.
We call those numbers ‘whole numbers’.
We use whole numbers when we count things, and counting things is very important in math.
The system that we count things with is based on the number 10. In fact, it’s called ‘Base 10’,
and it uses what we call ‘powers of 10’ as the groups or building blocks that we count with.
It’s probably no surprise that our first building block is ‘1’,
and we can get bigger building blocks just by multiplying by 10.
So our next building block is ‘1’ times 10 or 10.
And the one after that is 10 times 10 or 100.
And the one after that is 100 times 10 which is 1,000.
I could keep on multiplying by 10 to get bigger and bigger building block,
but I think I’m going to stop at 1,000 for now.
So what am I going to do with all these powers of 10 ?
Well, like I said, they’re the building blocks of our counting system, so I’m going to count with them!
But before I do that, I should mention two more things that our number system needs besides these building blocks.
And those two things are ‘digits’ and ‘number places’.
Digits are just the 10 different symbols we use for counting… you know: 0,1,2,3,4,5,6,7,8, and 9
Number places are like little imaginary boxes that can hold just one digit at a time,
and they’re used like ‘counters’… you know… to count things.
We use one number place for each of our building blocks, to count how many of each of them we have.
For example, let’s say I have ‘2’ hundreds, well then I just put a ‘2’ in the number place that counts how many hundreds I have.
And let’s say I have ‘5’ tens, then I put a ‘5’ in the number place that counts tens.
And I also have ‘3’ ones, so I put a ‘3’ in the number place that counts ones.
Each number place is named for the building block it counts.
For example, this number place is called the ‘ones place’ because it counts ones.
This is the ‘tens place’ because it counts tens,
the ‘hundreds place’ counts hundreds, and so on…
Of course, most of you aren’t used to seeing numbers written like this,
so let’s re-arrange our number places to make it look more like we are used to.
There… now you can see that the ‘2’ in the hundreds place,
the ‘5’ in the tens place,
and the ‘3’ in the ones place all combine to make 253.
Oh… and most of the time, we don’t actually show the number places… they’re invisible.
There… that’s better.
Okay, so that’s how we use our building blocks, number places and digits to write any whole number we want to.
But this video is about fractions, right?
So how are we going to write fractions with this system?
A lot of fractions have values smaller than one, but right now, our smallest building block is ‘1’.
It looks like we’re going to need some smaller building blocks.
You remember how we got our other ‘Base 10’ building blocks.
We started with ‘1’ and kept multiplying by 10 to get bigger and bigger building blocks.
Well… to get smaller building blocks, all we have to do is start with ‘1’ and keep dividing by 10.
Wait a minute? …divide a smaller number like ‘1’ by a bigger number like 10 ?
Well that sounds like a fraction to me!
Exactly!… If we do that, the building block we get will be a fraction,
and its value will be one part out of ten (or one-tenth),
and we can use that to write values that are smaller than one.
A good way to see how this new building block fits in with the other ones is to look at a number line.
Oh look! …a number line!
And it goes from zero to ten, and it show our first two building blocks: 1 and 10
If we take ’10’ and divide it into ten equal parts, each of those parts is equal to ‘1’.
And if we take ‘1’ and divide it into ten equal parts, then each of those parts is a tenth.
Now the number line shows three building blocks: 10, 1 and a tenth.
A tenth is a small number, but we can get even smaller building blocks if we keep dividing by 10.
So let’s take one-tenth and divide it into 10 equal parts.
This even smaller fraction is called a hundredth …which is one out of a hundred!
And since it’s ‘1’ over 100, it would take 100 of them to equal ‘1’.
We can see this by taking all 10 of our tenths and dividing each of them into 10 parts.
Now if you count up all these smaller parts, we’ll find that there are 100 of them.
We could keep on dividing with our number line to get smaller and smaller building blocks,
but I’ll show you an even easier way to find them.
Let’s list the building blocks we have so far.
Notice that on this side of the ‘1’, the first building block is 10,
and on the other side of the ‘1’, the first building block is 1 OVER 10.
And back on this side… our next building block is 100,
and on the other side side it’s 1 OVER 100.
Well then… if the next building block on this side is 1,000,
what do you suppose the next building block on the other side will be?
Yep… you guessed it! 1 OVER 1,000 (or a thousandth).
Of course, I could keep on going in either direction getting powers of 10
that are bigger and bigger whole numbers
or smaller and smaller fractions,
but I think these are enough for now.
Alright… now that we have all these new smaller or ‘fractional’ building blocks,
it means that we can count smaller and smaller parts of things.
But to do that, we’re going to need a new number place for each of them.
This will be called the ‘tenths place’ because it counts tenths.
And this will be the ‘hundredths place’ because it counts hundredths,
And this will be the ‘thousandths place’ because it counts thousandths.
To see how the new number places work, let’s bring back our last example digits:
2 hundreds, 5 tens and 3 ones,
but this time let’s add ‘6’ tenths by putting a ‘6’ in the tenths place,
and ‘4’ hundredths by putting a ‘4’ in the hundredths place.
Now let’s put our number places back like we are used to seeing them.
Uh-oh… It looks like there’s a problem!
This number looks like twenty-five thousand, three-hundred and sixty four!
But we only added tiny little fractions to our 253… it can’t be that big!
Here’s our problem… we can’t tell which number place is which because they all look the same.
What we need is a kind of a marker that will help us tell them apart.
And that marker is called a ‘decimal point’.
The decimal point is just a dot that we always put right here between the ones place and the tenths place.
That way, we always know that the ones place is on the left of the decimal point, and the tenths place is on the right.
There… that’s better! Now our number reads: 253 point 64.
That’s how you read the decimal point when you get to it… you just say “point”.
So the decimal point is really just a separator between the number places that count whole numbers on this side,
from the number places that count fractions on this side.
These amounts are greater than ‘1’.
And these amounts are smaller than ‘1’.
We call numbers that use a decimal point, “decimal numbers”, or “decimals” for short.
So that’s how decimal numbers work, and they’re important
because we use them to convert a fraction from a division problem to a regular number.
In the next section we’ll learn how to convert some special fractions into decimals.
But before that, let’s do a quick review.
If we take a fraction and do the division, we will get an answer.
And that answer is called the ‘value’ of the fraction.
Our number system is called ‘Base 10’.
It uses ‘powers of 10’ as building blocks for counting as well as 10 different digits.
Number places are like counters that hold one digit at a time
and help us count how many of each ‘Base 10’ building block a number is made of.
There are ‘Base 10’ building blocks like 1, 10, 100, and 1,000 that help us make really big numbers.
There are also very small ‘Base 10’ building blocks (fractions that are smaller than ‘1’) that help us make really small numbers.
These building blocks have names like tenths, hundredths, and thousandths.
The decimal point is a separator that goes between the number places that count values of ‘1’ or greater,
from the number places that count values smaller than ‘1’.
To make sure you understand how decimals work and how they relate to fractions, be sure to do those exercises!
…bet ya didn’t see that coming, did ya!
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