In this lesson, we’re going to learn how our basic number system works

and we’re going to learn about an important concept called Place Value.

The number system that we use in math is called ‘base 10’, because is uses ten different symbols for counting.

Math could use other systems that are based on a different number (like ‘base 2’ or ‘base 8’),

but I’ll give you ten guesses as to why the number ten is such a popular choice.

The ten symbols that we use are called ‘digits’ and they look like this:

zero, one, two, three, four, five, six, seven, eight and nine.

At first glance, you might think that’s only nine digits, but remember,

the zero counts as one of the digits also.

To see how our numbers system uses these digits to represent amounts,

let’s pretend that we have an apple orchard full of apple trees,

and each of these trees is loaded with big, juicy, red apples

that we need to pick and then count for our records.

[crunch]

We’re going to use something called a ‘number place’ to count.

The best way to understand a number place is to imagine that it’s like a small box

that’s only big enough to hold one digit at a time.

As we count, we’ll change the digit that’s in the number place to match how many apples we’ve picked.

For example, if we start with no apples at all, we put the digit ‘0’ in the number place because zero means ‘none’.

But then, as the apples start coming in from the orchard, we begin to count…

one, two, three, four, five, six, seven, eight and nine.

Okay, now we’ve got nine apples, but we’ve also got a problem.

We’ve already run out of digits to count with.

The highest digit we have is a ‘9’, but there are a LOT more apples left to count.

What will we do?

The solution is to use groups to help us count.

If we pick just one more apple, we’ll have ten, right?

So let’s combine those ten apples into a single group.

So… how many apples do we have? Ten!

BUT… how many GROUPS of ten apples do we have? Ah… just ONE!

Does that help us with our lack of digits problem?

It sure does, IF we use another number place!

Instead of using this new number place to count up individual apples

one at a time like we did with the first number place.

We’re going to use it to count apples TEN at a time.

In other words, we’ll use it to keep track of how many groups of ten apples that we’ve picked.

For example, if we’ve picked only one group of ten, then we’ll put the digit ‘1’ in that number place.

If we’ve picked two groups of ten, then we’ll put the digit ‘2’ in that number place,

and if we’ve picked three groups of ten, then we’ll put the digit ‘3’ in that number place. And so on.

Do you see what’s happening?

Because the new number place is being used to count GROUPS of ten,

it’s allowing us to re-use our original ten digits, but this time they are able to count (or represent) bigger amounts.

Since this new number place is for counting groups of ten, we’re going to name it ‘the tens place’.

And we’ll name our original number place, ‘the ones place’ because we used it to count things one at a time.

And here’s the really important thing… we’re not going to use the new number place instead of the old one…

we’re going to use it along side of the old one so that we have one number place for counting by ones

and another number place for counting by tens.

Using these two number places together lets us represent amounts that are in-between the groups of ten.

For example, if we’ve already picked thirty apples, then there will be a ‘3’ in the tens place

because we have three groups of ten.

But there will be a ‘0’ in the ones place, because there are no individual apples left over.

But… if we have picked thirty-two apples, then there will be a ‘3’ in the tens place

and a ‘2’ in the ones place to represent the two individual apples that are not in the groups of ten.

In fact, using only our ten digits and these two number places, we can count all the way from zero up to ninety-nine.

At ninety-nine, both of our number places are maxed out with the highest digits and we won’t be able to count any higher,

UNLESS... we get another number place!

If we’ve picked ninety-nine apples and then we pick just one more, we’ll have exactly one-hundred apples.

And if we make a group from those one-hundred apples,

we can use this new number place to count how many groups of one-hundred we’ve picked.

That means that we can re-use the same ten digits AGAIN in this new number place to count how many groups of one-hundred we have.

And you guessed it… it’s called ‘the hundreds place’ because we use it to count groups of a hundred.

Are you starting to see how our ‘base 10’ number system works?

It uses different number places to represent the different sized groups that we use to count.

And the digits in those number places tell us how many of each group we have.

The digit in the ones place tells us how many ones we have.

The digit in the tens place tells us how many groups of ten we have.

And the digit in the hundreds place tells us how many groups of one-hundred we have.

And have you noticed that each time we got a new number place to count larger groups,

we placed it to the LEFT of the previous number place.

That’s important because number places are always arranged in the exact same order.

Starting with the ones place, as you move to the left, the number places represent larger and larger amounts.

And did you also noticed that each new number place represents groups that are

exactly ten times bigger than the previous number place?

Ten is ten times bigger than one and one-hundred is ten times bigger than ten.

That’s really important because it helps us see the pattern for bigger number places.

It helps us to see that the next number place will count groups of ten times one-hundred, which is one-thousand.

That’s why it’s called ‘the thousands place’.

And the next number place will count groups ten times bigger than that! It’s the ten-thousands place!

And the number places keep on going like that.

Next is the hundred-thousands place.

Then… the millions place.

Then… ten-millions.

Then… one-hundred-millions.

Then… billions! And so on…

Oh, and you may notice that when we get a lot of number places next to each other like this,

it’s a little hard to quickly recognize which place is which.

That’s why many countries use some kind of separator every three places to make them easier to keep track of.

For example, in the U.S. we use a comma every three number places to make it easier

to identify things like the thousands place, or the millions place.

Seeing all these number places together helps you understand what we mean by ‘place value’.

In a multi-digit number, the number PLACE that a digit is in, determines it’s VALUE.

Even though we only have ten digits, each digit can stand for different amounts depending on the place that it occupies.

If the digit ‘5’ is in the ones place, it just means five.

But, if a ‘5’ is in the tens place, then it means fifty,

and if a ‘5’ is in the hundreds place, it means five-hundred.

And it’s the same for bigger number places.

A ‘5’ in the hundred-thousands place means five-hundred-thousand,

and a ‘5’ in the billions place means five-billion!

See how a digit’s place effects its value?

Of course, when we work with numbers in math, most of the time the number places are invisible.

But the underlying pattern is always the same.

Oh, and because the number places are invisible, in certain cases

you’ll need to use zeros to make it clear what number you’re talking about.

To see what I mean, imagine that this ‘5’ is in the hundreds place to represent five-hundred,

but if you make the number places invisible, then it just looks like five and not five-hundred.

So… to make sure people know you mean FIVE-HUNDRED, you need a ‘5’ in the hundreds place,

a ‘0’ in the tens place,

and a ‘0’ in the ones place.

Now you can tell that the ‘5’ is in the hundreds place and it means five-hundred.

Okay… now a great way to see place value in action with some actual numbers is to expand them

to show that they’re really combinations of different groups.

When we do this, it’s called writing a number in ‘Expanded Form’.

For example, we can expand 324 to be 300, 20 and 4,

because the ‘3’ is in the hundreds place and means three-hundred,

the ‘2’ is in the tens place and means twenty,

and the ‘4’ is in the ones place so it just means four.

So 324 in expanded form is the combination of those amounts:

three-hundred plus twenty plus four.

Let’s try writing another number in expanded form: 6,715

We can expand this into

six-thousand, (because the ‘6’ is in the thousands place)

plus seven-hundred, (because the ‘7’ is in the hundreds place)

plus ten, (because the ‘1’ is in the tens place)

and five, (because the ‘5’ is in the ones place)

So the expanded form is six-thousand plus seven-hundred plus ten plus five.

Alright… so do you see how our ‘base 10’ number system works?

Number places are used to count different sized groups.

Each group is ten times bigger than the next,

and the digits in the number places tell us how many of each group we have.

The tricky part is that the number places are invisible,

so you have to know how they work behind the scenes in order to make sense of multi-digit numbers.

How do you like them apples?

The exercises for this section will help you practice so that you get used to how place value works,

which is super important if you want to be successful in math.

As always, thanks for watching Math Antics and I’ll see ya next time.

[Clank!]

That gives me an idea...!

...I could make pies out of these!!

Learn more at www.mathantics.com