Large Whole Numbers: Place Values and Estimating

When we started counting things thousands of years ago, we quickly realized that numbers can get very big very fast, whether you're counting grains of sand or stars in the night sky.

Therefore, we couldn't give every possible number its own unique symbol.

We needed some kind of system that uses a small collection of symbols over and over to represent every numerical value imaginable.

That's why we came up with the notion of place values.

Say you're again counting apples.

You get 1, 2, 3, all the way up to 9.

Then when you hit 10, this becomes a two-digit number.

That's because we place a 1 in the tens place, while the units place goes back to 0 and starts again.

After it gets to 9 again, the tens place goes to a 2, indicating that we've reached 10 twice, while the units place goes back to 0.

This continues through 30, 40, 50, and when we reach 99, we've maxed out both the units place and the tens place, so we have to add the hundreds place.

We place a 1 here, indicating that we have reached 100, while the other two places go back to 0.

This process can continue ad infinitum.

Every time all the digits reach 9, we add a new digit to the left, a 1, while all the others go back to 0.

Thousands place, ten thousands, hundred thousands, millions, and so forth.

This is a brilliant system because it allows us to represent any number imaginable using the same 10 digits from 0 to 9, just in combinations that demonstrate various magnitudes.

The place values where these digits can be found indicate how big or small the number is.

So a 5 and a 6 and an 8 can come together to make 568, which is just five 100s plus six 10s plus eight ones.

If we see a 1 by itself, we know it is referring to a singular object, but when the 1 is sitting in the millions place, it takes on a whole new significance.

Just a few symbols can therefore represent any whole number that can be conceived of.

We can take advantage of this fact to make estimates.

We do this when we make a guess as to how many people are at a party, how many jelly beans are in a jar, or how long we've been waiting for the check at a restaurant.

If we know it's been more than five minutes but certainly less than 15, then 10 minutes is a good estimate because anywhere from five to 15 minutes when rounding to the tens place would give us 10 minutes.

This means that the 1 in the tens place is quite certain, while the 0 in the units place is uncertain.

That's what makes it an estimate, which is a kind of measurement.

We can do the same with the people at the party.

When we started counting things thousands of years ago, we quickly realized that numbers can get very big very fast, whether you're counting grains of sand or stars in the night sky.

Therefore, we couldn't give every possible number its own unique symbol.

We needed some kind of system that uses a small collection of symbols over and over to represent every numerical value imaginable.

That's why we came up with the notion of place values.

Say you're again counting apples.

You get 1, 2, 3, all the way up to 9.

Then when you hit 10, this becomes a two-digit number.

That's because we place a 1 in the tens place, while the units place goes back to 0 and starts again.

After it gets to 9 again, the tens place goes to a 2, indicating that we've reached 10 twice, while the units place goes back to 0.

This continues through 30, 40, 50, and when we reach 99, we've maxed out both the units place and the tens place, so we have to add the hundreds place.

We place a 1 here, indicating that we have reached 100, while the other two places go back to 0.

This process can continue ad infinitum.

Every time all the digits reach 9, we add a new digit to the left, a 1, while all the others go back to 0.

Thousands place, ten thousands, hundred thousands, millions, and so forth.

This is a brilliant system because it allows us to represent any number imaginable using the same 10 digits from 0 to 9, just in combinations that demonstrate various magnitudes.

The place values where these digits can be found indicate how big or small the number is.

So a 5 and a 6 and an 8 can come together to make 568, which is just five 100s plus six 10s plus eight ones.

If we see a 1 by itself, we know it is referring to a singular object, but when the 1 is sitting in the millions place, it takes on a whole new significance.

Just a few symbols can therefore represent any whole number that can be conceived of.

We can take advantage of this fact to make estimates.

We do this when we make a guess as to how many people are at a party, how many jelly beans are in a jar, or how long we've been waiting for the check at a restaurant.

If we know it's been more than five minutes but certainly less than 15, then 10 minutes is a good estimate because anywhere from five to 15 minutes when rounding to the tens place would give us 10 minutes.

This means that the 1 in the tens place is quite certain, while the 0 in the units place is uncertain.

That's what makes it an estimate, which is a kind of measurement.

We can do the same with the people at the party.