Binary & Number Representation

You’ll be able to

Explain why computers represent all data using bits (binary)
Convert numbers between binary and decimal
Relate the number of bits to the range of values that can be represented

Everything is bits

At the lowest level a computer stores and moves information as bits — binary digits, each either 0 or 1, physically an "off" or "on" state. Numbers, text, images, and sound are all represented as sequences of bits; the same 0s and 1s mean different things depending on how the program interprets them. A group of 8 bits is a byte. Binary works for the same reason our familiar decimal (base-10) system works, except each place is worth a power of 2 instead of a power of 10.

Place value in base 2

In binary, reading right to left, the place values are 1, 2, 4, 8, 16, 32, 64, 128, … — each place worth twice the one before it (2⁰, 2¹, 2², …). To find a binary number’s decimal value, add up the place values wherever there is a 1. For example, 1011 has 1s in the 8, 2, and 1 places: 8 + 2 + 1 = 11. Every whole number has exactly one binary representation, just as it has one decimal representation.

Bits and range: powers of two

The number of bits available limits how many different values you can represent. With n bits you can make 2ⁿ distinct patterns, representing the unsigned integers 0 through 2ⁿ − 1. Four bits give 2⁴ = 16 patterns (0–15); eight bits give 2⁸ = 256 patterns (0–255). This is why a value can overflow: if a number grows past the largest pattern the bits can hold, it cannot be represented and the result is wrong.

Range of n bits

patterns = 2ⁿ · largest unsigned value = 2ⁿ − 1

n bits produce 2ⁿ distinct patterns, representing the integers 0 through 2ⁿ − 1. Each added bit doubles the number of representable values.

Worked example

Convert the decimal number 45 to 8-bit binary.

1.Write the 8-bit place values: 128, 64, 32, 16, 8, 4, 2, 1.
2.Find the largest place value that fits in 45: 32 fits (45 − 32 = 13). Place a 1 in the 32s place.
3.From the remaining 13: 16 is too big (0), 8 fits (13 − 8 = 5), 4 fits (5 − 4 = 1), 2 is too big (0), 1 fits (1 − 1 = 0).
4.Place 1s in the 32, 8, 4, and 1 places and 0s elsewhere: 0 0 1 0 1 1 0 1.
5.Check: 32 + 8 + 4 + 1 = 45. ✓

Answer: 45 in 8-bit binary is 00101101 (32 + 8 + 4 + 1 = 45).

Checkpoint

What is the decimal value of the binary number `1101`?

Tip

Memorize the low powers of two: 1, 2, 4, 8, 16, 32, 64, 128, 256. Almost every binary conversion and range question on the exam comes straight from this list.

Checkpoint

A game stores a player’s score in a single unsigned byte (8 bits). What is the largest score it can represent, and what happens if the score should reach 256?

On the exam

Watch for overflow: with a fixed number of bits, the representable range is 0 to 2ⁿ − 1. A value past the top cannot be stored, so the exam expects you to say it overflows — not that it rounds or is stored fine.

Answer the 2 checkpoints as you read.