Place value is how we put digits together to create numbers just like we put together letters to create words. For example: "123" is one hundred twenty three but "231" is two hundred thirty one. These numbers are different even though they use the same digits. Just like the word "part" and "trap" are different words even though they use the same letters. In other words, the order of the digits is important.
Another way we could write the number "123" is as "100 + 20 + 3". Notice those zeroes in there that weren't in there before? That's because each digit - 1, 2, and 3 - has a place value that is replacing all the digits to its right with zero in order to represent its value at that place.
Place value is based on our base 10 number system - otherwise known as the decimal number system. What this means is that each digit is multiplied by an incrementally increasing power of 10. So the rightmost digit is multiplied by 100, the digit to the left of that digit is multiplied by 101, the digit to the left of the previous digit is multiplied by 102, and so on. What this looks like in expanded form can be explored in the table below.
Even though we can write increasing larger numbers, sometimes we have to communicate these numbers with words. Specifically, we have to say the name of the number out loud. In order to do that, each of these place values had to be given a name to refer to each number specifically. This system is a little weird, but there is a method to the madness. Specifically, three digits are grouped together and given an overall name, with each digit having a sub-name based on the very first grouping's naming. The first group of three digits - which make up the sub-naming system - are called the "hundreds" place, the "tens" place, and the "ones" place. For example: the number "123" has "1" in the hundred's place, "2" in the ten's place, and "3" in the one's place.
This base naming is then part of each grouping of three to the left of that and provided an additional name. The next grouping of three is referred to as the thousand's place. For example: the number "456,123" has "4" in the hundred-thousand's place, "5" in the ten-thousand's place, "6" one-thousand's place (or simply thousand's place), "1" in the hundred's space, "2" ten's space, and "3" in the one's place. The next grouping of three digits is the million's place, the next group after that the billion's place, and this continues. See the chart below to explore the relationship between powers of 10 and the names of each digit.
| trillions | billions | millions | thousands | ones | ||||||||||
| 1014 | 1013 | 1012 | 1011 | 1010 | 109 | 108 | 107 | 106 | 105 | 104 | 103 | 102 | 101 | 100 |
hundred trillions | ten trillions | trillions | hundred billions | ten billions | billions | hundred millions | ten millions | millions | hundred thousands | ten thousands | thousands | Hundreds | Tens | Ones |
Using the chart above, we can develop a way of converting a number written like "123" into words like "one hundred twenty-three". What makes writing numbers a little less simple than expected is that we have shorthand for numbers in the one's and ten's places. Specifically, we don't say "50" as "five ten's" but rather "fifty". And at a deeper level, "53" is written as "fifty-three" and not "five tens, three ones". For the one's place, we simply drop the word "one's". Another rule is that we don't say the name of a place value that has a value of 0. In the example of "50" we don't say "five tens and zero one's" - it's just "50". In the case of "101" we don't say "one hundred, zero ten's, and one one", we say "one hundred and one" completely skipping over the ten's place. The final rule we'll introduce is that each 3-digit grouping name - million, billion, trillion - get said after we've said the name of that grouping's sub-name. For example: "456,123" is "four-hundred fifty-six thousand, one-hundred twenty-three". Another example: "789,456,123" is "seven-hundred eighty-nine million, four-hundred fifty-six thousand, one-hundred twenty-three." The diagram below is an interactive way of exploring how to write any number.
An interesting fact is that there are names for numbers with up to 102 digits. After that we only have names for a couple special numbers. The largest named number is 1010100 - known as a googolplex. Fun, random fact: the company Google named it's headquarter's the "googleplex" as a play on words whereby googolplex is a number and the word "complex" is a name for a cluster of buildings.
The usage of place value allows us to communicate very large numbers with some very simple rules. Place value allows us to describe how any individual digit can represent that number multiplied by increasingly larger powers of the number 10. Not only can we write the number in digits, but we also have a system whereby we can communicate these numbers with words. But, the craziest part to me is that we can write a number with only digits and understand what that number means without actually having a name for that number. That, my friend, is the power of mathematical notation and place value!