Theory
One plus one is... not two
Chirag writes on the whiteboard:
1
+ 1
----
"In this shop's machines," he says, "the answer is not 2. There IS no 2."
Binary has only the symbols 0 and 1, so when a column's total reaches two, something else must happen, the same something that happens in decimal when a column reaches ten. You already know this move; you have used it since primary school.
Theory
The carry you already know
In decimal, 7 + 5 = 12: you write 2 and carry 1, because 12 crossed the base (ten). Binary does the identical thing one step earlier: 1 + 1 = two, which crosses the base (two), so you write 0 and carry 1. Written in binary, two IS "10". Same old carry rule, just a smaller base triggering it sooner.
At a glance
The complete rulebook
| Addition | Result | Subtraction | Result |
|---|---|---|---|
| 0 + 0 | 0 | 0 - 0 | 0 |
| 0 + 1 or 1 + 0 | 1 | 1 - 0 | 1 |
| 1 + 1 | 0, carry 1 | 1 - 1 | 0 |
| 1 + 1 + 1 | 1, carry 1 | 0 - 1 | 1, borrow 1 |
Theory
Worked: 1011 + 110
Align right, add column by column with carries on top:
carry: 1 1 1
1 0 1 1 (= 11)
+ 1 1 0 (= 6)
------------------
1 0 0 0 1 (= 17)
Column walk (right to left): 1+0 = 1. Then 1+1 = 0 carry 1. Then 0+1+carry = 0 carry 1. Then 1+carry = 0 carry 1. Carry falls out front: 1.
Verify in decimal: 11 + 6 = 17 ✓. Never skip this line.
Theory
Worked: 1010 - 11
1 0 1 0 (= 10)
- 1 1 (= 3)
------------------
1 1 1 (= 7)
Right column: 0 - 1 cannot go, borrow from the next column: 10 - 1 = 1. Next column gave away its 1, so it is 0 - 1: borrow again, 10 - 1 = 1. Next: 0 - 0 (after lending) = 1... careful, walk it slowly on paper.
Verify: 10 - 3 = 7 ✓. Borrows chain exactly like decimal borrowing across zeros (like 100 - 1).
Think first
Your turn: 101 + 11
Add 101 + 11 on paper, carries written above each column. Then verify in decimal before revealing.
Show the answer
carry: 1 1
1 0 1 (= 5)
+ 1 1 (= 3)
--------------
1 0 0 0 (= 8)
Right: 1+1 = 0 carry 1. Middle: 0+1+1 = 0 carry 1. Left: 1+carry = 0 carry 1, which falls out front. 1000₂ = 8 ✓. If you wrote 112 anywhere, the base police would like a word.
Quiz
In binary, what is 1 + 1?
- 10
- 2
- 11
- 1
Show the answer
10
Two crosses binary's base, so it is written as 10 (one-zero, meaning one two and zero ones), i.e. write 0, carry 1. The symbol 2 simply does not exist in binary; 11 would be three. This tiny question appears in some form in nearly every paper.
Watch out
Where marks leak
A digit 2 anywhere in a binary answer: instant giveaway of a base slip. Dropped chained carries: when 1+1+1 happens, the sum is 1 AND the carry moves on; write carries above columns, always. Borrow bookkeeping: after lending, the lender column shrinks by 1; track it in writing, not memory. And end every problem with the decimal verification line.
Theory
You just did the ALU's job
Remember the ALU in the block diagram? Column-by-column binary addition with carries is literally what its adder circuits do, billions of times per second. Subtraction in real hardware uses a clever trick (two's complement) you will meet later; the rules you learned today are the foundation it stands on.
Summary
Key takeaways
- Addition rules: 0+0=0, 0+1=1, 1+1=0 carry 1, 1+1+1=1 carry 1.
- Subtraction rules: 1-0=1, 1-1=0, 0-1=1 with a borrow from the next column.
- Write carries above columns and track borrows on paper.
- The symbol 2 never appears in binary work.
- Always verify by converting operands and answer to decimal.
- Memory hook: one plus one is one-zero.