13 Other Bases
In the 1←3 system, three dots in one box is worth one dot in the box one spot to the left. This gives a new picture:
Each dot in the second box from the left is worth three ones. Each dot in the third box is worth three 3’s, which is nine, and so on.
Example
We said that the 1←3 code for fifteen is 120. We see that this is correct because:
Problem 8
Answer these questions about the 1←3 system.
- What label should go on the box to the left of the 9 box?
- What would be the value of a box two spots to the left of the 9 box?
- What number has 1←3 code 21002?
- What is the 1←3 code for two hundred dots?
In the 1←4 system, four dots in one box are worth one dot in the box one place to the left.
Problem 9
Answer these questions about the 1←4 system.
- What is the value of each box in the picture above?
- What is the 1←4 code for twenty-nine dots?
- What number has 1←4 code 132?
Problem 10
In the 1←10 system, ten dots in one box are worth one dot in the box one place to the left.
- Draw a picture of the 1←10 and label the first four boxes with their values.
- What is the 1←10 code for eight thousand four hundred and twenty-two?
- What number has 1←10 code 95,753?
- When we write the number 7,842, what does the “7” represent?
The “4” is four groups of what value?
The “8” is eight groups of what value?
The “2” is two groups of what value? - Why do you think we use the 1 ← 10 system for writing numbers?
Definition
Recall that numbers written in the 1←2 system are called binary or base two numbers.
Numbers written in the 1←3 system are called base three numbers.
Numbers written in the 1←4 system are called base four numbers.
Numbers written in the 1←10 system are called base ten numbers.
In general, numbers written in the 1←b system are called base b numbers.
In a base b number system, each place represents a power of b, which means for some whole number n. Remember this means b multiplied by itself n times:
- The right-most place is the units or ones place. (Why is this a power of b?)
- The second spot is the “b” place. (In base ten, it’s the tens place.)
- The third spot is the “” place. (In base ten, that’s the hundreds place. Note that .)
- The fourth spot is the “” place. (In base ten, that’s the thousands place, since .)
- And so on.
Notation
Whenever we’re dealing with numbers written in different bases, we use a subscript to indicate the base so that there can be no confusion. So:
- is a base three number (read it as “one-zero-two base three”). This is the base three code for the number eleven.
- is a base four number (read it as “two-two-two base four”). This is the base four code for the number forty-two.
- is a base ten number. (It’s ok to say “fifty-four thousand three hundred and twenty-one.” Why?)
If the base is not written, we assume it’s base ten.
Remember: when you see the subscript, you are seeing the code for some number of dots.
Think / Pair / Share
- Find the number of dots represented by each of these:
- Represent nine d