Binary and Hex

Twos Complement

Two’s complement is the standard way computers represent signed integers in binary.
Positive numbers are stored as usual in binary, while negative numbers are represented by inverting all bits of the positive value and adding 1.

This system has several advantages:

Addition and subtraction use the same hardware for signed and unsigned values
There is only one zero
The leftmost bit acts as the sign bit and supports sign extension

Because of these properties, two’s complement is used by virtually all modern CPUs for integer arithmetic.

Example:

Represent −18 as an 8-bit two’s complement value

Write the magnitude in binary

\[18_{10} = 00010010_2\]

Invert the bits

00010010  <- what we started with
11101101  <- inverted/flipped

Add 1 to the previous result

Add 1 to get the two’s complement representation:
```
 11101101
+       1
---------
 11101110
```

The final result is 11101110

Hexadecimal

Example

Convert the binary value 10110110 to hexadecimal.

Group bits into nibbles (4 bits)

Starting from the right, group the binary digits into sets of four:
```
1011 0110
```
Convert each nibble to hex
```
Binary: 1011  0110
Hex:       B     6
```

The final value is B6 in hex.