Chapter 2 · The basis of a number system
A few symbols. So many numbers.
How do we write a quantity when we have used every single digit? Grouping lets us reuse symbols in new positions.
Optional read-aloud uses your browser’s voice.
Question
Why does a base-ten digit stop at nine? What does 10 show that a single digit cannot?
A small cue
Think of ten individual objects becoming one group of ten. What must we record about the group and the objects left over?
Grouping gives each place its value
In a positional system, the base tells us how many units of one place form one unit of the next larger place. In base ten, ten units make one ten. Ten tens make one hundred.
| Base | Digit symbols | Whole place values |
|---|---|---|
| 2 | 0, 1 | 1, 2, 4, 8, … |
| 8 | 0 to 7 | 1, 8, 64, 512, … |
| 10 | 0 to 9 | 1, 10, 100, 1000, … |
| 16 | 0 to 9 and A to F | 1, 16, 256, 4096, … |
Why 10 does not always mean ten
In base b, the digits run from 0 to b − 1. The next count is written 10: one group of b and zero individual units. So 10 in base two has decimal value two; 10 in base eight has decimal value eight.
Try it
What is 10 in base two worth in decimal?
Choose an answer when you are ready.
Reflect
Could you explain why base eight needs seven beads per digit position and an empty count for zero?
A note about time
Sixty seconds per minute and sixty minutes per hour are grouped measurements. A week is a seven-day cycle. These conventions should not all be described as ordinary uniform-base numeral systems.