If you have “31” potatoes, how many do you have?
Really. It’s not a trick question. So give it a try — what do you think? How many potatoes do you have when you have “31”?
When I wrote 31, you likely assumed that I meant “three groups of ten, plus one more.” You understood that 31 means that quantity. It was like seeing the written string of letters “pie” and thinking of a round, fruit-filled pastry dessert.
But you may also know that the letters “pie” don’t always refer to a dessert. If I were writing in Spanish, “pie” would refer to someone’s foot.
The string of letters P-I-E has two meanings. Each makes sense in its own context.
The same thing is true of 31. In most situations, 31 means “three groups of ten, plus one more.” That’s because nowadays, we use what is called the base ten system.
We all relate to base ten as “normal.” But it is important to understand what is going on when we go from 9 to 10.
To understand what base ten really does, imagine that I’m a potato farmer, with, say, 1436 potatoes to ship to my friends outside of Idaho. I have special crates to put my potatoes in. Each potato crate holds exactly ten potatoes: no more, no fewer.
I use my potatoes to fill as many crates as possible. Any single potatoes left over, after all the crates are full, are put on the side. Those single potatoes are represented by the rightmost digit (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9). So, with 1436 potatoes, and crates that hold ten each, I will have 6 potatoes left over.
You probably know how this pattern continues. The second-to-rightmost digit works like the rightmost digit, but on a bigger scale. I pack all the ten-potato crates into big sacks. Each sack holds ten crates – that’s ten crates, not ten potatoes. I fill as many sacks as possible, and then count how many crates are left over. 1436 potatoes leaves me with 3 crates not in sacks along with the 6 potatoes not in crates.
And of course, the third digit to the right works the same way. Not only am I shipping my friends some of my mountain-grown potatoes, I actually have an order from a distributor on the East Coast. I load the sacks into trucks, and each truck holds ten sacks. The third-to-rightmost digit represents the number of sacks left when I have filled as many trucks as possible. If there are 1436 potatoes, that means that once the trucks are filled, there are 4 sacks, 3 crates and 6 potatoes left.
You might have noticed a pattern here. Each crate holds ten potatoes. Each sack holds ten groups of ten potatoes: that’s ten times ten potatoes, or one hundred potatoes. Each truck holds ten times ten times ten potatoes: one thousand potatoes. Every step up in size – that is, every digit further to the left – holds ten times as many potatoes as the last one. If, say, there were a barge that held ten trucks, it would hold ten times ten times ten times ten potatoes: ten thousand potatoes.
This is why we can also write 1436 in a way similar to something you may remember from grade school: “1 group of one thousand potatoes, 4 groups of one hundred potatoes, 3 groups of ten potatoes, and 6 single potatoes.” It is the same as writing, “1 group of (ten times ten times ten) potatoes, 4 groups of (ten times ten) potatoes, 3 groups of ten potatoes, and 6 single potatoes.” Likely none of this is revolutionary: you probably realized it already, on some level.
But with this understanding now explicit, other base systems suddenly become interesting. To jump to another base, there is only one thing you need to do. Change “ten” to another number.
Imagine you go to a new Base Sixteen potato farm, and are shocked to see larger crates! Base sixteen represents how potatoes would be sorted at that farm. Instead of crates that hold ten potatoes, sacks that hold ten crates, and so on, the crates hold sixteen potatoes, the sacks hold sixteen crates, and so on. To the people who store those potatoes, 1436 means “1 group of (sixteen times sixteen times sixteen) potatoes, 4 groups of (sixteen times sixteen) potatoes, 3 groups of sixteen potatoes, and 6 single potatoes.” That’s a lot more potatoes than the “1436” from my farm.
In base sixteen, the set of digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 is not enough. Eleven, for example, can’t be written in base sixteen as a combination of those digits. Usually, people use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F to represent zero through fifteen in base sixteen. So, 6A means “6 groups of sixteen, plus eleven more.” Base sixteen is also called hexadecimal. It is used a lot in computer science, including as a way to represent colors in HTML. (I’m a fan of #8904B1 and #00FFBF.) This very website, like many others, uses hexadecimal notation to specify the font color and background color.
You may be wondering how all of this helps you count to one thousand and twenty-three on your fingers.
Believe it or not, it IS possible to count that high. All you need is the notion of base and a little creativity.
I’ll show you how in the next post.