Explain why computers use binary, describe place value in binary and hexadecimal, and state why hexadecimal is used as shorthand for binary

What this dot point is asking

SEAB wants you to explain why computers use the binary number system, to describe place value in binary (base 2) and hexadecimal (base 16), and to say why hexadecimal is used as a shorthand for binary. The central idea is that a computer is built from switches with only two states, so two digits are enough, and that hexadecimal simply groups those binary digits into a more readable form.

The answer

Why computers use binary

A computer is made of millions of tiny electronic switches called transistors. Each switch is either on or off, with no reliable in-between state. The binary number system has exactly two digits, $0$ and $1$, so we can let $0$ mean off and $1$ mean on. Every piece of data, whether a number, a letter, a sound or an image, is ultimately stored as a pattern of these two values.

Using only two states makes the hardware simple and reliable: it is easy to tell apart a high voltage from a low voltage, even with a little electrical noise.

Bits and bytes

A single binary digit is called a bit. A group of $8$ bits is called a byte. One byte can store $256$ different patterns, from $00000000$ to $11111111$, which is enough to represent one text character or a number from $0$ to $255$.

Place value in binary

Binary uses positional place value, exactly like denary, but each column is a power of two instead of a power of ten:

So the 4-bit number $1101_2$ has bits in the $8$, $4$ and $1$ columns:

Place value in hexadecimal

Hexadecimal uses base 16, so each column is a power of sixteen, and it needs sixteen digits. The digits run $0$ to $9$ and then $\text{A}, \text{B}, \text{C}, \text{D}, \text{E}, \text{F}$ for the values ten to fifteen:

Why hexadecimal is shorthand for binary

Because $16 = 2^4$, one hexadecimal digit represents exactly four binary bits, called a nibble. This makes hex a compact way to write binary: an 8-bit byte that needs eight binary digits needs only two hex digits.

binary:  1010 1111
hex:        A    F   -> AF

Examples in context

Example 1. Web colours. A colour on a web page such as #FF8800 is three pairs of hex digits for the red, green and blue intensities. Each pair is one byte, so FF is $255$ (full intensity) and 00 is $0$ (none). Designers read these directly because hex packs each byte into two digits.

Example 2. File sizes. When a phone reports a photo as taking up several megabytes, it is counting bytes, and each byte is eight bits. Understanding that a byte holds $256$ patterns is the first step to understanding why a higher-quality image, with more bits per pixel, takes more storage.

Try this

Q1. State what a bit and a byte are, and how they are related. [2 marks]

• Cue. A bit is a single binary digit ($0$ or $1$); a byte is a group of eight bits.

Q2. Give the denary value of $11100_2$. [2 marks]

• Cue. Place values $16, 8, 4, 2, 1$, so $16 + 8 + 4 = 28$.

Q3. Explain why one hexadecimal digit can represent any 4-bit binary value. [2 marks]

• Cue. Four bits give $2^4 = 16$ patterns, and hexadecimal has exactly $16$ digits ($0$ to $\text{F}$), so each pattern maps to one hex digit.