2.1 Binary Numbers

Why binary?

Computers store and process data using binary numbers, a base-2 system with only two digits: 0 and 1. Each digit is called a bit. Binary works well for computers because digital circuits can easily represent two states — on/off, high/low voltage — making it a reliable way to encode information.

Converting binary to decimal

Each bit position represents a power of 2, starting from 2^0 on the right. To convert binary to decimal, multiply each bit by its place value and sum the results. For example, 1011 = (1×8)+(0×4)+(1×2)+(1×1) = 11 in decimal.

Limits and overflow

A binary number with n bits can represent 2^n distinct values, ranging from 0 to 2^n − 1. If a calculation produces a value too large for the available bits, it causes overflow, where the extra bits are lost and the stored value becomes incorrect.

Key terms

Binary number system

A base-2 number system that uses only the digits 0 and 1 to represent data.

Bit

The smallest unit of data in computing, representing a single binary digit (0 or 1).

Converting binary to decimal

Multiply each bit by its place value (a power of 2) and sum the results.

Number of values with n bits

2^n distinct values can be represented, ranging from 0 to 2^n − 1.

Overflow

An error that occurs when a computed value is too large to be represented with the available number of bits, causing incorrect results.

Try an AP-style question

What is the binary number 1011 in decimal?

1. 10
2. 1011
3. 13
4. 11