Grouping in base 60
No landmark is ever used 60 times or more.
Seventy sixties
The Mesopotamians used base 60. Their landmark numbers were 1, 60, 60² = 3600, … Look at this number: (1) × 3600 + (70) × 60 + 2. It has 70 sixties.
What would you do with 70 sixties?
What this lesson covers
The idea
When a number is grouped into powers of 60 (the sexagesimal system), no power of 60 occurs 60 or more times, because 60 of them group into the next power; so the counts 1–59 suffice for every power.
Seventy sixties
The Mesopotamians used base 60. Their landmark numbers were 1, 60, 60² = 3600, … Look at this number: (1) × 3600 + (70) × 60 + 2. It has 70 sixties.
What would you do with 70 sixties?
- Group 60 of them into one 3600
- Leave it, 70 is fine
- Break them into ones
Group sixty into one
Tap 60 → 1 on a count that is 60 or more. It makes one of the next landmark. Watch the value at the bottom.
Counts from 1 to 59
When a number is grouped into powers of 60 (the sexagesimal system), no power of 60 occurs 60 or more times, because 60 of them group into the next power; so the counts 1–59 suffice for every power.
(1) × 3600 + (70) × 60 + 2 = (2) × 3600 + (10) × 60 + 2 = 2 ; 10 ; 2
Notes
When a number is grouped into powers of 60 (the sexagesimal system), no power of 60 occurs 60 or more times, because 60 of them group into the next power; so the counts 1–59 suffice for every power.
Check yourself
In (1) × 3600 + (70) × 60 + 2, after grouping 60 of the 60s, how many 3600s are there?
Answer: 2
We had 1 and made 1 more: 2 sixty-squares. The number is (2) × 3600 + (10) × 60 + 2.
After one group is made from 70 sixties, how many 60s are left over?
Answer: 10
70 − 60 = 10, so 10 sixties are left over.
A number has 125 ones in base 60. Group 60 ones into one 60 as many times as possible. How many ones are left?
Answer: 5
125 = 2 × 60 + 5, so we get 2 sixties and 5 ones left.
Why does a base-60 numeral never need a count of 60 or more?
- Because 60 of any landmark number group into 1 of the next landmark — correct. Yes! So every count stays between 1 and 59 (or is absent).
- Because 60 is a very large number to count. Large counts are no problem for counting. The reason is that 60 of one landmark always make one of the next.
- Because we run out of symbols after 59. The symbols do not run out. Groups of 60 are simply made into the next landmark, so no count reaches 60.