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Addition in base five works just like addition in base ten. The only difference is the size of the groups we trade. Every digit sits in a place, and the value of that place depends on the base. In base 5 we only use the digits 0, 1, 2, 3 and 4.
… thousands hundreds tens ones
Each place is 10 times the place on its right:
So when we say “carry 1” from the tens place, we are really carrying 1 hundred (because 10 tens make 1 hundred).
… 125s 25s fives ones
Each place is 5 times the place on its right:
So when we say “carry 1” from the fives place, we are really carrying 1 twenty-five (because 5 fives make 1 twenty-five).
The process is the same. The only difference is when we carry: in base 10 we carry when a column reaches 10; in base 5 we carry when a column reaches 5.
| thousands | hundreds | tens | ones | |
|---|---|---|---|---|
| Carry | 1 | 1 | 1 | |
| 9 | 7 | 8 | ||
| + | 8 | 5 | 6 | |
| Sum | 1 | 8 | 3 | 4 |
8 + 6 = 14 = 10 + 4.
Write 4 in the ones place.
The 10 ones = 1 ten → carry a 1 to the tens place.
1 ten + 7 tens + 5 tens = 13 tens = 10 tens + 3 tens.
Write 3 in the tens place.
The 10 tens = 1 hundred → carry a 1 to the hundreds place.
1 hundred + 9 hundreds + 8 hundreds = 18 hundreds = 10 hundreds + 8 hundreds.
Write 8 in the hundreds place.
The 10 hundreds = 1 thousand → carry a 1 to the thousands place.
Bring down the 1. Result: 1834.
| 125s | 25s | fives | ones | |
|---|---|---|---|---|
| Carry | 1 | 1 | 1 | |
| 4 | 3 | 2 | ||
| + | 3 | 4 | 4 | |
| Sum | 1 | 3 | 3 | 1 |
2 + 4 = 6 ones = 1 five + 1 one.
Write 1 in the ones place.
The 5 ones = 1 five → carry a 1 to the fives place.
1 five + 3 fives + 4 fives = 8 fives = 5 fives + 3 fives.
Write 3 in the fives place.
The 5 fives = 1 twenty-five → carry a 1 to the 25s place.
1 twenty-five + 4 twenty-fives + 3 twenty-fives = 8 twenty-fives
= 5 twenty-fives + 3 twenty-fives.
Write 3 in the 25s place.
The 5 twenty-fives = 1 one-hundred-twenty-five → carry a 1 to the 125s place.
Bring down the 1. Result: 13315
In base 10, ten is written 10 because 10 ones = 1 ten + 0 ones. In base 5, five is written 105 for the same reason. There is no digit “5” in base 5:
2 + 3 = 5 ones = 1 five + 0 ones.
Write 0 in the ones place.
Carry the 1 five to the fives place (written simply as a 1).
This table shows every sum of two base 5 digits, written in base 5. The pink cells reach 5 or more, so they produce a carry: the first digit is the carry and the second digit is what you write.
| + | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 | 10 |
| 2 | 2 | 3 | 4 | 10 | 11 |
| 3 | 3 | 4 | 10 | 11 | 12 |
| 4 | 4 | 10 | 11 | 12 | 13 |
When you add three or more numbers, a column can total enough to produce a carry of 2, 3 or more (two or three of the next higher place). The method stays the same.
| thousands | hundreds | tens | ones | |
|---|---|---|---|---|
| Carry | 2 | 2 | 2 | |
| 5 | 8 | 6 | ||
| + | 6 | 9 | 7 | |
| + | 8 | 5 | 9 | |
| Sum | 2 | 1 | 4 | 2 |
6 + 7 + 9 = 22 = 20 + 2.
Write 2 in the ones place.
The 20 ones = 2 tens → carry 2 to the tens place.
2 + 8 + 9 + 5 = 24 tens = 20 tens + 4 tens.
Write 4 in the tens place.
The 20 tens = 2 hundreds → carry 2 to the hundreds place.
2 + 5 + 6 + 8 = 21 hundreds = 20 hundreds + 1 hundred.
Write 1 in the hundreds place.
The 20 hundreds = 2 thousands → carry 2 to the thousands place.
Bring down 2. Result: 2142.
| 125s | 25s | fives | ones | |
|---|---|---|---|---|
| Carry | 2 | 2 | 2 | |
| 4 | 4 | 4 | ||
| + | 4 | 4 | 4 | |
| + | 4 | 4 | 4 | |
| Sum | 2 | 4 | 4 | 2 |
4 + 4 + 4 = 12 ones = 10 ones + 2 ones = 2 fives + 2 ones.
Write 2 in the ones place.
Carry 2 to the fives place.
2 + 4 + 4 + 4 = 14 fives = 10 fives + 4 fives
= 2 twenty-fives + 4 fives.
Write 4 in the fives place.
Carry 2 to the 25s place.
2 + 4 + 4 + 4 = 14 twenty-fives
= 2 one-hundred-twenty-fives + 4 twenty-fives.
Write 4 in the 25s place.
Carry 2 to the 125s place.
Bring down 2. Result: 24425
Add the column in ordinary numbers, then ask: how many full fives fit, and what is left over? The number of fives is the carry. The leftover (0–4) is the digit you write.
Example: 4345 + 3435 + 4445 + 3245
| 125s | 25s | fives | ones | |
|---|---|---|---|---|
| Carry | 3 | 3 | 3 | |
| 4 | 3 | 4 | ||
| + | 3 | 4 | 3 | |
| + | 4 | 4 | 4 | |
| + | 3 | 2 | 4 | |
| Sum | 3 | 2 | 1 | 0 |
Total 15 → three fives + 0 left → write 0, carry 3
Total 16 → three fives + 1 left → write 1, carry 3
Total 17 → three fives + 2 left → write 2, carry 3
Just the carry → write 3
When a column has a large total (from the numbers plus any incoming carry):
13 = 5 + 5 + 3 → write 3, carry 2.
20 = 5 + 5 + 5 + 5 + 0 → write 0, carry 4.
0–4 → carry 0 · 5–9 → carry 1 · 10–14 → carry 2 · 15–19 → carry 3 · 20–24 → carry 4
Add the base 5 numbers shown. Fill in the carry digits (except the ones place — there is never a carry into the ones place) and the sum digits. Start from the right. Remember: only the digits 0–4 can appear in the sum.
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