Addition in base two is similar to addition in base ten. Although there is a difference, it is not that hard to add in base 2. Every digit sits in a place. The value of that place depends on the base.
… 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).
… eights fours twos ones
Each place is 2 times the place on its right:
So when we say “carry 1” from the twos place, we are really carrying 1 four (because 2 twos make 1 four).
Connection: 100 is not a mystery — it is simply 10 tens. In the same way, 4 is simply 2 twos, and 8 is 2 fours. Carrying is just trading a full group for one unit in the next higher place.
What is the value of 11012 in base 10?
13. 1 eight + 1 four + 0 twos + 1 one = 8 + 4 + 0 + 1 = 13.
The process is the same. The only difference is when we carry: in base 10 we carry when a column reaches 10; in base 2 we carry when a column reaches 2.
| thousands | hundreds | tens | ones | |
|---|---|---|---|---|
| Carry | 1 | 1 | 1 | |
| 9 | 7 | 8 | ||
| + | 8 | 5 | 6 | |
| Sum | 1 | 8 | 3 | 4 |
Ones place
8 + 6 = 14 = 10 + 4.
Write 4 in the ones place.
The 10 ones = 1 ten → carry a 1 to the tens place.
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.
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.
Thousands place
Bring down the 1. Result: 1834.
| eights | fours | twos | ones | |
|---|---|---|---|---|
| Carry | 1 | 1 | 1 | |
| 0 | 1 | 1 | ||
| + | 1 | 0 | 1 | |
| Sum | 1 | 0 | 0 | 0 |
Ones place
1 + 1 = 2 ones = 1 two + 0 ones.
Write 0 in the ones place.
The 1 two is carried as a 1 to the twos place.
Twos place
1 two + 1 two + 0 twos = 2 twos = 1 four + 0 twos.
Write 0 in the twos place.
The 1 four is carried as a 1 to the fours place.
Fours place
1 four + 0 fours + 1 four = 2 fours = 1 eight + 0 fours.
Write 0 in the fours place.
The 1 eight is carried as a 1 to the eights place.
Eights place
Bring down the 1. Result: 10002
Key idea: What we really carry is 1 ten / 1 hundred / 1 thousand (or 1 two / 1 four / 1 eight). In practice we just write the digit 1. The column we put it in already tells us its value.
Writing 2 in the answer. In base 2 the only digits are 0 and 1, so 1 + 1 is never written as 2: it is 1 two + 0 ones, so you write 0 and carry 1.
Add 1102 + 0112.
10012. Ones: 0 + 1 = 1. Twos: 1 + 1 = 2 → write 0, carry 1. Fours: 1 + 1 + 0 = 2 → write 0, carry 1. Eights: 1. Check: 6 + 3 = 9 = 8 + 1.
In base 10, ten is written 10 because 10 ones = 1 ten + 0 ones. In base 2, two is written 102 for the same reason:
Ones place
1 + 1 = 2 ones = 1 two + 0 ones.
Write 0 in the ones place.
Carry the 1 two to the twos place (written simply as a 1).
In short: 1 + 1 = 2 ones = 1×2 + 0×1 → written 102.
Same pattern as 5 + 5 = 10 ones = 1×10 + 0×1 → written 10 in base 10.
10112 + 1102 = 100012
How is the number 3 written in binary?
112. 3 = 1 two + 1 one.
A column can total enough to produce a carry of 2 or 3 (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 |
Ones place
6 + 7 + 9 = 22 = 20 + 2.
Write 2 in the ones place.
The 20 ones = 2 tens → carry 2 to the tens place.
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.
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.
Thousands place
Bring down 2. Result: 2142.
| sixteens | eights | fours | twos | ones | |
|---|---|---|---|---|---|
| Carry | 1 | 2 | 2 | 1 | |
| 1 | 1 | 1 | |||
| + | 1 | 1 | 1 | ||
| + | 1 | 1 | 1 | ||
| Sum | 1 | 0 | 1 | 0 | 1 |
Ones place (1s)
1 + 1 + 1 = 3 ones = 1 two + 1 one.
Write 1 in the ones place.
The 1 two is carried as a 1 to the twos place.
Twos place (2s)
1 two + 1 two + 1 two + 1 two = 4 twos = 4 twos + 0 twos = 2 fours + 0 twos.
Write 0 in the twos place. Carry 2 to the fours place.
Fours place (4s)
2 fours + 1 four + 1 four + 1 four = 5 fours = 4 fours + 1 four = 2 eights + 1 four.
Write 1 in the fours place. Carry 2 to the eights place.
Eights place (8s)
2 eights = 1 sixteen + 0 eights.
Write 0 in the eights place.
The 1 sixteen is carried as a 1 to the sixteens place.
Sixteens place (16s)
Bring down 1. Result: 101012
Same rule in both bases: Count the column total (including any carry coming in) → write the leftover after making full groups of 10 (base 10) or 2 (base 2) → carry the number of full groups to the next place (you may carry 1, 2, or 3).
A binary column has four 1s in it (including the carry). What do you write, and what do you carry?
Write 0, carry 2. Four 1s make 2 pairs and nothing is left over.
Pair the 1s in each column. Each pair → carry 1 left. Leftover 1 → write 1. No leftover → write 0.
Example: 11112 + 11012 + 10112 + 11112 + 1112
| thirty-twos | sixteens | eights | fours | twos | ones | |
|---|---|---|---|---|---|---|
| Carry | 1 | 3 | 3 | 3 | 2 | |
| 1 | 1 | 1 | 1 | |||
| + | 1 | 1 | 0 | 1 | ||
| + | 1 | 0 | 1 | 1 | ||
| + | 1 | 1 | 1 | 1 | ||
| + | 1 | 1 | 1 | |||
| Sum | 1 | 1 | 1 | 1 | 0 | 1 |
1s: 1+1+1+1+1
Five 1s → two pairs + leftover → write 1, carry 2
2s: 1+0+1+1+1 + carry 2
Six 1s → three pairs → write 0, carry 3
4s: 1+1+0+1+1 + carry 3
Seven 1s → three pairs + leftover → write 1, carry 3
8s: 1+1+1+1 + carry 3
Seven 1s → three pairs + leftover → write 1, carry 3
16s: carry 3
Three 1s → one pair + leftover → write 1, carry 1
32s: carry 1
Just the carry → write 1
Fast rule: Pair the 1s. Each pair → carry 1. Leftover 1 → write 1. No leftover → write 0.
Why 6 twos = 3 fours: 2 twos make 1 four, so 6 twos make 3 fours.
Result: 1111012 (15 + 13 + 11 + 15 + 7 = 61 = 32 + 16 + 8 + 4 + 1)
When a column contains several 1s (from the numbers plus any incoming carry):
Six 1s
1+1+1+1+1+1 = 6 = 2+2+2+0 → write 0, carry three 1s.
Five 1s
1+1+1+1+1 = 5 = 2+2+1 → write 1, carry two 1s.
Same idea as base 10: write the ones digit, carry the number of tens. In binary: write the leftover 0 or 1, carry the number of twos.
A binary column has seven 1s (including the carry). What do you write, and what do you carry?
Write 1, carry 3. Seven 1s make 3 pairs with 1 left over.
Add the binary numbers shown. Fill in the carry bits (except the ones place — there is never a carry into the ones place) and the sum bits. Start from the right. “2 numbers” means adding two binary numbers; “3 numbers” means adding three; and so on.
Now check what you learned. Every answer is explained, and each time you play you get new questions.
Quiz complete!