Binary Fractions

Just like denary decimals, binary fractions use positions to the right of a decimal point to represent parts of a whole. Each position represents a fraction with powers of 2 in the denominator.

The pattern is straightforward: after the decimal point, positions represent 1/2, 1/4, 1/8, 1/16, and so on. For example, 0.1 in binary equals 1/2 (0.5 in denary), whilst 0.01 equals 1/4 (0.25 in denary).

Quick Tip: Remember that each position after the decimal point halves in value, just like whole number positions double going left!

Converting becomes easier once you recognise these fraction patterns. The binary 0.101 represents 1/2 + 1/8, which equals 0.5 + 0.125 = 0.625 in denary.

Sign and Magnitude vs Two's Complement

Computers need ways to represent negative numbers, and there are two main methods you'll encounter. Sign and magnitude is the simpler approach - it uses the leftmost bit as a sign indicator.

In sign and magnitude, the most significant bit acts like a positive or negative sign. A 0 means positive, whilst a 1 means negative. So +16 and -16 look identical except for that first bit.

The problem? Arithmetic operations become tricky with this system. That's where two's complement comes in - a more sophisticated method where the leftmost bit represents an actual negative value rather than just a sign.

Key Point: Two's complement makes computer arithmetic much easier because addition and subtraction work the same way for positive and negative numbers!

Two's complement uses the leftmost bit to represent a negative number $like -128 in an 8-bit system$, making calculations more straightforward for processors.

Converting to Two's Complement

Converting positive numbers to negative using two's complement follows a specific three-step process that's actually quite manageable once you practise it.

Start with your positive binary number written out completely. Then, working from right to left, copy each digit exactly until you reach the first 1 - copy that 1 as well.

From that point onwards, flip every remaining bit. All 0s become 1s, and all 1s become 0s. The result is your negative number in two's complement form.

Memory Trick: Think "copy to the first 1, then flip the rest" - this method works every time and saves you from making mistakes!

This system makes computer arithmetic work seamlessly. Unlike sign and magnitude, you can add and subtract two's complement numbers using the same circuits, which is why virtually all modern computers use this method.