Counting in base 2.

Step 1 of 7

How many values in how many bits

The screen shows a picture of a sunset, and in the computer it is only numbers. Each pixel stores three numbers, one for its red light, one for green and one for blue, and each of those is stored in Bits per channel bits. The number of bits decides how many different levels each channel can take, from off to full.

Each channel is now stored in 3 bits. Predict, and then Look inside will enlarge a patch of the picture, split each pixel into its three lights, and stand a ruler with a tick for every level beside the columns.

Step 2 of 7

Each bit doubles

An extra bit does not add two more values; it doubles them all, because every old pattern appears once with a 0 in the new bit and once with a 1. So the number of values in $n$⁠ bits is a power of two:

$2^1 = 2,\quad 2^2 = 4,\quad 2^3 = 8,\quad 2^4 = 16,\quad \ldots$$2^1 =$$2,\quad 2^2 =$$4,\quad 2^3 =$$8,\quad 2^4 =$$16,\quad \ldots$⁠

That is exponential growth: it starts slowly and very soon becomes huge. Predict what happens when the bits double.

Step 3 of 7

By the number of levels

The ruler beside the columns now shows the number of levels at each depth. Change Bits per channel and watch the bands in the sky crowd together as the levels multiply, until to the eye they are gone.

Step 4 of 7

A byte

Eight bits are a byte. It is the size computers measure memory and files in, and in true colour each channel is one byte. A byte has $2^8 = 256$⁠ values. What is the largest of them?

Step 5 of 7

Reading a byte

To read a byte, write the eight place values over it, from left to right:

$128 \quad 64 \quad 32 \quad 16 \quad 8 \quad 4 \quad 2 \quad 1$⁠

and add the ones with a 1 under them. To keep a long byte readable, it is usually written in two groups of four, like $0100\,1101_2$⁠.

Step 6 of 7

Writing a byte

The other way, from base ten to base 2, starts from the largest place. Check whether $128$⁠ fits in the number: if it does, write 1 and take it away; if not, write 0. Move to $64$⁠ with what is left, and so on down to the 1s place. At the end you should be left with $0$⁠.

Step 7 of 7

How many bits you need

Sometimes the question is the other way round: you have $N$⁠ things, and each needs a number of its own. How many bits do you need? Look for the smallest power of 2 that reaches $N$⁠. That is the logarithm in base 2, $\log_2 N$⁠, rounded up, and it is why it is so useful in computer science: each time $N$⁠ doubles, you need just one more bit.