Intuition The one idea behind everythingA computer has only two symbols, 0 and 1, and no wire that means "minus". Two's complement is the single trick of giving the top bit a negative weight so that the same adder circuit handles positive numbers, negative numbers, addition and subtraction — all at once.
Before you can use that trick, you have to be fluent in the little symbols the parent note throws around: bits, place-values, the bar x ˉ ⊕ C
A bit is one switch that is either off (0) or on (1). That's the entire alphabet of a computer. There is no 2, no -, no . — only these two states.
Think of a light switch on a wall. Down = 0, up = 1. A number is a whole row of these switches. Everything else on this page is about agreeing on what a row of switches means .
Why the topic needs it: two's complement is a rule for reading a row of switches . If you don't picture the switch, the later symbols (b n − 1 x ˉ
b i i b n − 1 b n − 2 … b 1 b 0 b subscript i position , counted from the right starting at 0. So b 0 b 1 b n − 1
Intuition Why start counting at 0, and why does position matter?Look at the figure: the rightmost switch sits over the label 2 0 2 1 i is the exponent of 2 that switch controls. Naming positions this way is what lets us later say "the top switch b n − 1
n n = 8 b n − 1 leftmost switch — the parent note calls it the most significant bit (MSB) and gives it the negative weight.b 0 rightmost switch — the least significant bit .
This is the foundation the whole topic sits on, so we build it slowly. See Binary number system for the unsigned home of this idea.
Definition Weight (place-value)Each switch is worth a fixed amount. In ordinary binary the switch at position i 2 i 2 i 2 0 = 1 , 2 1 = 2 , 2 2 = 4 , 2 3 = 8 , … on .
Intuition Why powers of two and not something else?With only two states per switch, each new switch on the left can be either off (adds nothing) or on (adds a fresh amount larger than everything to its right combined). Doubling each step is exactly the rule that makes "on = one bigger place" work with no overlaps. In the figure the on-switches (pale yellow) have their weights summed to give the value.
00000101
Only positions 2 0 4 + 1 = 5
Why the topic needs it: the parent's master formula is just this weighted sum, with one weight made negative .
Definition Sigma / summation∑ ∑ add up a list . The line ∑ i = 0 n − 2 b i 2 i i 0 n − 2 i b i ⋅ 2 i + + +
It's a for-loop over the switches . The bottom label (i = 0 n − 2 b 0 2 0 + b 1 2 1 + …
Why the topic needs it: the definition
V = − b n − 1 ⋅ 2 n − 1 + ∑ i = 0 n − 2 b i ⋅ 2 i ∑ except the top one)", then tacks on the top switch with a minus in front. Recognise the two pieces and the formula stops being scary.
Definition Signed weightingIn two's complement the top switch is worth − 2 n − 1 + 2 n − 1 (everything below) minus (the top switch's amount) .
Intuition Why one negative weight fixes everythingLook at the figure's number line. The lower switches can only build up to 2 n − 1 − 1 2 n − 1 flips you to the negative side . The single most-negative value − 2 n − 1
Top off, rest anything: values 0 2 n − 1 − 1
Top on: values − 2 n − 1 − 1
Together: [ − 2 n − 1 , 2 n − 1 − 1 ]
Why the topic needs it: this is the topic. Everything else (negation rule, discarding carries) is a consequence.
Definition Bitwise NOT, writtenx ˉ flip every switch : every 0 becomes 1 and every 1 becomes 0. This is the logic gate NOT applied to all switches at once. See One's complement , which is this flip with nothing added.
2 n − 1 2 n n one short of a clean 2 n
Why the topic needs it: the parent derives − x = x ˉ + 1
2 n A ≡ B ( mod 2 n ) A B 2 n n 2 n look identical , because the difference falls off the top and there's no switch to hold it. See Modular arithmetic .
Intuition The odometer / clock pictureThe figure is a circle of 2 n 0; counting down from 0 wraps to all-ones. So 2 n 0 same point , and 2 n − x − x we just walk the other way around the circle.
Why the topic needs it:
It explains why 2 n − x ≡ − x x ˉ + 1
It explains why you discard the carry-out : the carry is worth 2 n ≡ 0
⊕ ⊕ exclusive OR : the result is 1 when the two inputs differ , 0 when they match. So 1 ⊕ 0 = 1 1 ⊕ 1 = 0
Definition Carry into / out of the MSBWhen adding column by column (like a Full adder chain), each column can send a spillover 1 into the next column — a carry . C n − 1 into the top column; C n out of the top column. The subscript is just the position the carry is crossing.
Intuition Why compare these two carries?If the same amount of carry flows in and out of the top column, the sign came out honestly. If they disagree (C n − 1 ⊕ C n = 1 is signed overflow. Full detail lives in Overflow and carry flags .
Why the topic needs it: this is the only extra symbol needed for the overflow rule Overflow = C n − 1 ⊕ C n
Position and subscript b_i
Summation adds weighted bits
Identity x plus not x = 2^n minus 1
Negation rule not x plus 1
Test yourself — you're ready for the parent note when you can answer each without peeking.
What does a single bit represent, physically? One switch that is either off (0) or on (1) — the only alphabet a computer has.
In b n − 1 … b 0 b 0 b n − 1 b 0 b n − 1
What is the weight of the switch at position i 2 i
What does ∑ i = 0 n − 2 b i 2 i Add up the weighted value of every switch except the top one.
What is different about the MSB's weight in two's complement? It is − 2 n − 1 + 2 n − 1
What does the bar in x ˉ Flips every bit: 0↔1 (bitwise NOT).
What does x + x ˉ 2 n − 1 0 and one 1.
What does A ≡ B ( mod 2 n ) n A B 2 n
Why can you discard the carry-out in signed addition? It is worth 2 n 2 n ≡ 0 ( mod 2 n )
What does C n − 1 ⊕ C n = 1 Carry into the MSB differs from carry out of it → signed overflow.
