Base Ten

Mentor: Tell me what you know about base ten.

Student: I know about base ten blocks.

Mentor: Then tell me about those.

Student: You use them to show numbers. There are little cubes, those represent 1. Then there are long sticks that have ten of the cubes in them. Those represent 10. Then there are these squares that have ten of the sticks in them. They represent 100.

Mentor: That's exactly right. But how do you use them to show numbers?

Student: You take some sticks, let's use 4. That means that you have 40.

Mentor: How do you know it's 40?

Student: Because it's 4 groups of ten. That's 4 x 10 = 40. There are 40.

Mentor: So you have 40. Now what?

Student: Then you could add 4 of the cubes. If you put the 4 sticks with the 4 cubes, you get a number: 44.

Mentor: You've got it, exactly. To do that, you're using base 10. That just means that there will be 10 cubes in your sticks and 10 sticks in your square, and so on.

Student: But why do we need a fancy name for that? Everyone knows there are 10.

Mentor: Imagine that I had something I called "base two blocks". How many cubes would make up a stick then?

Student: I don't know. Probably 2.

Mentor: That's right! In base 2, there are 2 cubes in a stick, and 2 sticks in a square. Imagine that you had 3 cubes in base 10. How would you show this in base 2?

Student: Well, you'd have to put 2 of the cubes together to make a stick. Then you'd have a leftover cube.

Mentor: Right! What would you call that number you just showed me with base 2 blocks?

Student: It's 3.

Mentor: No, it's not 3. In base 2, 3 doesn't exist. Think about sticks and cubes again. Pretend like that stick is just like a base 10 stick. What would you call the number?

Student: In base 10, if you have 1 stick and 1 cube, you put the 1 for the stick in the ten's place, and the 1 for the cube in the one's place and you get 11. But it's not 11, so I don't know what to do.

Mentor: You do it exactly like that. In base 2, you count the sticks, and put that number in something we'll call the two's place. Then count the cubes and put that number in the one's place.

Student: But it's not eleven!

Mentor: It's not eleven in base 10, no. We know that it's 3. But in base 2, we write the number like this: 11. Do you understand?

Student: I think so. If we were writing the number 2 in base two, would we write it like this: 10?

Mentor: Exactly. Now you're ready to try more difficult numbers!