Binary Builder

Binary Builder 💡

A computer is built from switches that are either **on** or **off**. We write those two states as **1** and **0**, and that is **binary**: counting in base 2. Every number, letter, sound and image inside a computer is, underneath, just a pattern of 1s and 0s.

Bits, nibbles and bytes 📦

The units build up from a single binary digit: - A **bit** is one binary digit (0 or 1). - A **nibble** is 4 bits. - A **byte** is 8 bits. Each extra bit **doubles** the number of patterns you can make, so **n bits give 2 to the power n** different values.

How many patterns?

An interactive activity.

The place values 🔟

In an 8-bit binary number, each column has a **place value**, doubling from right to left: **128 64 32 16 8 4 2 1** To read a binary number, add up the place values wherever there is a 1. For example **0010 1101** = 32 + 8 + 4 + 1 = **45**.

Binary to denary

An interactive activity.

Denary to binary

Which is the number 37 written as 8-bit binary? (37 = 32 + 4 + 1.)

• 0010 0101
• 0010 1001
• 0011 0100
• 0100 1010

Adding binary ➕

Binary addition works column by column, right to left, with the four rules: - 0 + 0 = 0 - 0 + 1 = 1 - 1 + 1 = 10 (write 0, **carry 1**) - 1 + 1 + 1 = 11 (write 1, **carry 1**) So 0101 + 0011: the carries ripple left to give **1000** (that is 5 + 3 = 8).

Add the binary numbers

An interactive activity.

Overflow ⚠️

A fixed number of bits can only hold numbers up to a limit: 8 bits reach **255** (1111 1111). If a calculation needs a bigger number than the bits allow, the extra carry has nowhere to go. This is called **overflow**, and it makes the stored answer wrong. It is the reason a byte cannot store 300.

True of number bases

Select the TWO statements that are true.

• Binary uses only the digits 0 and 1
• Overflow happens when a result is too big for the available bits
• Binary is base 10
• 8 bits can represent 512 different values

Hexadecimal 🔠

Long binary numbers are hard for people to read, so we often use **hexadecimal** (base 16) as a shorthand. Its digits are 0-9, then **A, B, C, D, E, F** for 10 to 15. The neat part: **one hex digit is exactly four binary bits** (a nibble). So a byte is just two hex digits, for example 1111 1111 = **FF**.

Match each hex digit to its 4-bit binary

Hex to denary

An interactive activity.

Binary summary

Binary is base _____. With n bits you can make 2 to the power n values, so 8 bits give _____ values. Each hexadecimal digit stands for _____ binary bits. Working out 0110 + 0011 in binary gives _____.