Two's Complement and Fixed-Point Binary

This lesson covers how computers represent signed integers using two's complement and how fixed-point binary is used to represent fractional numbers. Both are required for the OCR H446 specification.

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Why Signed Numbers?

Computers need to represent both positive and negative integers. Since binary only has 0s and 1s, a convention is needed to distinguish between positive and negative values.

Approaches to Signed Representation

Method	Description	Used in Practice?
Sign and magnitude	The MSB is the sign (0 = positive, 1 = negative); remaining bits = magnitude	Rarely — has +0 and -0
One's complement	Negative = flip all bits of the positive value	Rarely — has +0 and -0
Two's complement	Negative = flip all bits and add 1	Yes — standard method

Two's complement is the standard because:

• There is only one representation of zero
• Addition and subtraction work using the same hardware circuit
• The range is slightly asymmetric but well-defined

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Two's Complement Representation

In an n-bit two's complement system:

• The most significant bit (MSB) represents the negative of its place value
• All other bits represent their normal positive place values

8-bit Two's Complement Place Values

-128	64	32	16	8	4	2	1
MSB (sign bit)							LSB

Range

For an n-bit two's complement number:

• Minimum value: -(2^(n-1))
• Maximum value: 2^(n-1) - 1

Bits	Minimum	Maximum
8	-128	+127
16	-32,768	+32,767
32	-2,147,483,648	+2,147,483,647

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Converting Denary to Two's Complement

Positive Numbers

If the number is positive, convert to binary as normal. The MSB will be 0.

Example: +45 in 8-bit two's complement = 00101101

Negative Numbers

1. Convert the magnitude (absolute value) to binary
2. Flip all bits (one's complement)
3. Add 1

Worked Example: Convert -73 to 8-bit Two's Complement

Step 1: 73 in binary = 01001001 Step 2: Flip all bits = 10110110 Step 3: Add 1 = 10110111

-73 in two's complement = 10110111

Verification

-128	64	32	16	8	4	2	1
1	0	1	1	0	1	1	1

-128 + 32 + 16 + 4 + 2 + 1 = -128 + 55 = -73. Correct.

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Converting Two's Complement to Denary

Method 1: Use the Place Values Directly

If the MSB is 0, the number is positive — convert as normal.

If the MSB is 1, the number is negative — add up all place values, remembering the MSB represents a negative value.

Worked Example: Convert 11100010 to Denary

-128	64	32	16	8	4	2	1
1	1	1	0	0	0	1	0

-128 + 64 + 32 + 2 = -30

Method 2: Reverse the Process

If the MSB is 1:

1. Flip all bits
2. Add 1
3. The result is the magnitude; the original was negative

11100010 -> flip -> 00011101 -> add 1 -> 00011110 = 30

So the original = -30

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Two's Complement Arithmetic

The beauty of two's complement is that addition works the same regardless of sign.

Example: 25 + (-10)

  25 = 00011001
 -10 = 11110110

  00011001
+ 11110110
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  00001111  = 15