# Numbers/Content/What Are Numbers/Bases

• Base

## Lesson

You learned in the last lesson how each place amount in a number is a group of ten. Why do people use groups of ten? Why not groups of 12, 8, or 60? Most people have ten fingers and ten toes, so it is easy to count in groups of ten.

A base is any number you use to make groups for counting. Math works with any number as a base.

## Example Questions

### Question 1:

What are the numbers one through ten using a base of three?

There are only three numerals in base three, so we select 0, 1, and 2.

The first two numbers are the same as in base ten.

• 1
• 2

When we get to three we must start filling the threes place.

• 10
• 11
• 12

Again at six we increase the threes place

• 20
• 21
• 22
• 100
• 101

### Question 2:

What is the number 265 in base five

One way to answer this is to write every number up to 265 in base five. This is easy, but it takes a long time. A faster way to do this is to think about groups of five. In base five the first place is the ones place, the second place is the fives place, the third place is the 25's place, and the fourth place is the 125's place.

Take two groups of 125 from 265 and you have 15 left. 15 is three groups of five. This means you put a two in the 125's place and a three in the fives place. The answer is $\displaystyle 2030$ .

### Question 3:

This question may be too hard if you do not know about adding. See the article Using Numbers, Adding and Subtracting for help.

What are the next binary (base 2) numbers in base 10?

• $\displaystyle 000011$
• $\displaystyle 000101$
• $\displaystyle 001000$
• $\displaystyle 001101$
• $\displaystyle 010101$
• $\displaystyle 100010$

The place amounts for base 2 are in this order:

• ones
• twos
• fours
• eights
• 16's
• 32's

To find the base 10 number find the place amount of each numeral and find the number you have all together.

• $\displaystyle 000011$
One and two together make $\displaystyle 3$ .
• $\displaystyle 000101$
One and four together make $\displaystyle 5$ .
• $\displaystyle 001000$
The fourth place is $\displaystyle 8$ .
• $\displaystyle 001101$
One and four and eight together make $\displaystyle 13$ .
• $\displaystyle 010101$
One and four and 16 together make $\displaystyle 21$ .
• $\displaystyle 100010$
Two and 32 together make $\displaystyle 34$ .