Converting From Base Ten

Student: I know how to convert numbers from base ten to base two, but are there other bases that I need to learn?

Mentor: Yes, there are, but you've already learned them.

Student: How is that possible? We've only talked about base ten and base two.

Mentor: But the same rules apply for all of the other number bases. Let's try one and see: 18 in base ten is what in base 5?

Student: Well, 15 ones is the same as 3 fives, right? That leaves 3 ones left. So it should be 33. Is that right?

Mentor: Yes, it is. See? You can do it. Now do you think you can take a number in base five and convert it to base ten?

Student: Let's try. What about the number 521. Let's convert that from base five to base ten.

Mentor: I don't think we can do that. The number 521 doesn't exist in base five. Think about it and see if you can explain why.

Student: I guess the 5 isn't right. In base two, 2 was written as 10. In base five, I guess 5 should be written as 10, too. But why do we call it base five if there aren't any fives?

Mentor: Good question. Think about how 5 is special in base five. Don't think about the number. Instead, picture five objects. What's special about a group that is that big?

Student: Well, when there are 5...

Mentor: No, we're not saying "5" anymore. That number doesn't exist in this base.

Student: When there are that many cubes, we put them together to make a stick, and we write it 10.

Mentor: Exactly. In base five, 5 is the number that makes up the next place value. In base ten, it's really easy to add and subtract with 10, isn't it? Do you think that is the case in base five?

Student: No, 5 is a lot harder to add and subtract with.

Mentor: But it's not 5 anymore, remember? Try to add this 1 stick of five cubes to 2 more cubes. How do you write that number in base five?

Student: Well, the 1 stick plus 2 cubes is written "12". Oh, I get it! You add just like you do in base ten, only the numbers stand for different amounts.

Mentor: Why do you think that is?

Student: When you're adding, you don't actually need to count the cubes on the stick. You just need to count the stick. So 1 stick (even if it has only 5 cubes) and 2 cubes is 12.

Mentor: Exactly right! Now keep that in mind when you're working with other numbers and bases. When you change bases, all you're really changing is how long each stick is.