Professor Dave here, let’s talk about place values.
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 one, two, three, all the way up to nine.
Then when you hit ten, this becomes a two-digit number.
That’s because we place a one in the tens place, while the units place goes back to
zero and starts again.
After it gets to nine again, the tens place goes to a two, indicating that we’ve reached
ten twice, while the units place goes back to zero.
This continues through thirty, forty, fifty, and when we reach ninety-nine, we’ve maxed
out both the units place and the tens place, so we have to add the hundreds place.
We place a one here, indicating that we have reached one hundred, while the other two places
go back to zero.
This process can continue ad infinitum.
Every time all the digits reach nine, we add a new digit to the left, a one, while all
the others go back to zero.
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 ten digits, from zero to nine, 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 five and a six and an eight can come together to make five hundred sixty eight,
which is just five one hundreds plus six tens plus eight ones.
If we see a one by itself, we know it is referring to a singular object, but when the one 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 jellybeans
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 fifteen, then ten
minutes is a good estimate, because anywhere from five to fifteen minutes, when rounding
to the tens place, would give us ten minutes.
This means that the one in the tens place is quite certain, while the zero 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.
We could count by groups of roughly ten and see that it’s definitely more than fifty,
but nowhere near two hundred, so one hundred becomes a reasonable estimate.
The one in the hundreds place is rather certain, while the tens and units places are uncertain.
The human mind has a phenomenal ability to approximate in this regard.
But now that we’ve learned how to represent any large number towards the limit of the
infinitely large, we have to understand how to also represent the infinitely small, or
the infinitesimal, so let’s learn about decimals next.