2.1 Binary Numbers

A bit (short for binary digit) is the smallest unit of data: a single 0 or 1. A byte is 8 bits. Computing devices are digital, which means that at the lowest level every value they store is a pattern of bits.

Bits are grouped to stand for bigger ideas: numbers, letters, colors, sounds. That's an abstraction, which means hiding the details you don't need so you can focus on the main idea. You think "the letter A" or "the color red," not "01000001."

The same bits can mean different things in different contexts. The byte 01000001 is the number 65 when read as a whole number, and it's the letter A in the common ASCII text code. A color is often stored as three bytes, one each for red, green and blue, each from 0 to 255. The program decides how to read the bits.

Data values can be stored in variables, in lists, or as fixed constants, and they can be passed into and returned from procedures.

Decimal (base 10) uses the digits 0 to 9. Binary (base 2) uses only 0 and 1. In both, a digit's position sets its value: each place is the base raised to a power, starting at power 0 on the far right.

In decimal, the places are 1, 10, 100, 1000 and so on. In binary, they're 1, 2, 4, 8, 16, 32, 64, 128, doubling each time you move left.

Binary to decimal: multiply each bit by its place value and add. For 1101: 8 + 4 + 0 + 1 = 13.

Decimal to binary: find the largest place value that fits, put a 1 there, subtract, and repeat with what's left; every place you skip gets a 0.

With n bits you can make 2ⁿ different patterns. If they stand for whole numbers starting at 0, the range is 0 to 2ⁿ − 1. So 4 bits hold 0 to 15, and 8 bits (one byte) hold 0 to 255.

Each extra bit doubles the number of values, it doesn't just add one. To give 100 students each a different ID, you need 7 bits, because 2⁶ = 64 is too few and 2⁷ = 128 is enough.

To compare binary numbers with the same number of digits, read from the left: the first place where they differ decides which is bigger. 1011 is larger than 1001 because they first differ in the 2s place.

Many programming languages store whole numbers in a fixed number of bits. That limits the range, and a result outside it causes an overflow error. If a program stores whole numbers in 4 bits, 15 + 1 doesn't fit, because 16 needs 5 bits.

Some languages avoid this by letting whole numbers grow as large as the computer's memory allows. The exam's pseudocode works this way, so overflow only comes up in questions about a fixed number of bits.

Real numbers (numbers with decimals) are also stored in a fixed number of bits, so many can only be approximated. That causes round-off errors. In many languages 0.1 + 0.2 comes out as 0.30000000000000004, not exactly 0.3. You won't be asked the exact ranges for real numbers.

Analog data changes smoothly over time instead of in separate steps: the pitch of a voice, the brightness across a sunset, the position of a runner. Computers can't store infinitely smooth values, so they approximate them.

Sampling means measuring the analog signal at regular intervals. Each measurement, called a sample, is stored as bits. Taking more samples per second, or using more bits per sample, makes the digital copy closer to the original but uses more storage. Representing analog data digitally is another example of abstraction.