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  • Hi, I’m Rob. Welcome to Math Antics.

  • In this lesson, were going to learn how our basic number system works

  • and were going to learn about an important concept called Place Value.

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

  • Math could use other systems that are based on a different number (likebase 2’ orbase 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 calleddigitsand 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]

  • Were going to use something called a ‘number placeto 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, well change the digit that’s in the number place to match how many apples weve picked.

  • For example, if we start with no apples at all, we put the digit ‘0’ in the number place because zero meansnone’.

  • 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 weve got nine apples, but weve also got a problem.

  • Weve 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, well have ten, right?

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

  • Sohow many apples do we have? Ten!

  • BUThow many GROUPS of ten apples do we have? Ahjust 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.

  • Were going to use it to count apples TEN at a time.

  • In other words, well use it to keep track of how many groups of ten apples that weve picked.

  • For example, if weve picked only one group of ten, then well put the digit ‘1’ in that number place.

  • If weve picked two groups of ten, then well put the digit ‘2’ in that number place,

  • and if weve picked three groups of ten, then well 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, were going to name itthe tens place’.

  • And well name our original number place, ‘the ones placebecause we used it to count things one at a time.

  • And here’s the really important thingwere not going to use the new number place instead of the old one

  • were 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 weve 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.

  • Butif 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 weve picked ninety-nine apples and then we pick just one more, well 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 weve 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 itit’s calledthe hundreds placebecause we use it to count groups of a hundred.

  • Are you starting to see how ourbase 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 calledthe 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.

  • Thenthe millions place.

  • Thenten-millions.

  • Thenone-hundred-millions.

  • Thenbillions! 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 byplace 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

  • youll need to use zeros to make it clear what number youre 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.

  • Soto 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.

  • Okaynow a great way to see place value in action with some actual numbers is to expand them

  • to show that theyre really combinations of different groups.

  • When we do this, it’s called writing a number inExpanded 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.

  • Alrightso do you see how ourbase 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

Hi, I’m Rob. Welcome to Math Antics.

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A2 初級

数学アンチックス - 位相値 (Math Antics - Place Value)

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    Yassion Liu に公開 2021 年 01 月 14 日
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