Two's Complement Binary Computer Science โ Student Guide
Two's Complement Binary
How computers store and work with negative numbers
Computers store everything as 1s and 0s. Positive whole numbers are straightforward in binary โ
but what about negative numbers? You can't write a minus sign in binary! The standard solution
used in virtually every modern computer is two's complement, a clever system that lets addition
and subtraction work with the same circuitry for both positive and negative numbers.
1. The Problem with Negative Numbers
With 8 bits we can store the unsigned values 0โ255. If we want to store negative numbers too, we
need to use one bit to indicate the sign. Two's complement reserves the most significant bit
(MSB) โ the leftmost bit โ as the sign bit: 0 = positive, 1 = negative.
MSB (sign) Bit 6 Bit 5 Bit 4 Bit 3 Bit 2 Bit 1 Bit 0 (LSB)
01011010
Red = sign bit. This pattern starts with 0, so it is positive. With 8-bit two's complement the range is
โ128 to +127.
Bits Range of values
4-bit โ8 to +7
8-bit โ128 to +127
16-bit โ32,768 to +32,767
32-bit โ2,147,483,648 to +2,147,483,647
Formula: For n bits: range is โ2nโ1 to +2nโ1 โ 1.
2. How to Find the Two's Complement
To convert a positive number into its negative two's complement equivalent, follow two steps. We'll
convert +19 into โ19 using 8 bits.
1
Write the positive number in binary, then flip every bit (one's complement)
+19 in 8-bit binary: 00010011
Flip every bit (0โ1): 11101100 โ this is the one's complement
2
Add 1 to the result
11101100 + 00000001 = 11101101
So โ19 in 8-bit two's complement is 11101101. Notice the MSB is 1 โ confirming it's
negative. โ
๐ก Shortcut method
