Place value system
The same symbol means more when it sits further left.
Same symbols, different slots
The Mesopotamians dropped the separate symbols for 1, 60, 3600 … Instead they wrote only the counts, and where each count stood told which landmark number it belonged to.
The symbols 2 5 are written once. What could tell us whether they mean 25, 250 or 2500?
What this lesson covers
The idea
A number system with a base that uses the position of each symbol to decide which landmark number it stands for is a positional or place value system; it represents unending numbers with finitely many symbols.
Same symbols, different slots
The Mesopotamians dropped the separate symbols for 1, 60, 3600 … Instead they wrote only the counts, and where each count stood told which landmark number it belonged to.
The symbols 2 5 are written once. What could tell us whether they mean 25, 250 or 2500?
- Their position among the landmark slots
- Their colour
- Nothing could tell us
Slide the digits
Each slot is a landmark number. Slide the digits 2 5 left and right. An empty slot counts as nothing. Watch the value.
Position decides the landmark
A number system with a base that uses the position of each symbol to decide which landmark number it stands for is a positional or place value system; it represents unending numbers with finitely many symbols.
The digit 2 stood for 2 × 10, then 2 × 100, then 2 × 1000. Only its position changed.
Notes
A number system with a base that uses the position of each symbol to decide which landmark number it stands for is a positional or place value system; it represents unending numbers with finitely many symbols.
Check yourself
The digit 2 is in the hundreds slot and the digit 5 in the tens slot. What number is it?
Answer: 250
2 × 100 + 5 × 10 = 200 + 50 = 250.
The digit 3 is in the thousands slot and 8 in the hundreds slot. The other slots are empty. What number is it?
Answer: 3800
3 × 1000 + 8 × 100 = 3000 + 800 = 3800.
In base 60 the digits 2 ; 5 ; 30 stand in the slots 3600, 60, 1. What number is it?
Answer: 7530
2 × 3600 + 5 × 60 + 30 = 7200 + 300 + 30 = 7530.
What is the great advantage of a place value system?
- It writes unending numbers using only finitely many symbols — correct. Yes! The position tells the landmark, so no new symbols are needed.
- Every symbol always has the same value. In a place value system the same symbol has different values in different positions.
- It needs a new symbol for every landmark number. That was the problem before. With place value we do not need a symbol for each landmark number.