Bits on switches.

Step 1 of 5

One bit

A computer is built from billions of tiny switches, transistors, and each can be in only one of two states: conducting or not, high voltage or low. The two states are called $0$⁠ and $1$⁠. A digit like that, with only two values, is a bit, short for binary digit.

The board has two switches, A and B, and each is a bit. The gates and lights after them are the next Unit’s subject; for now, look only at the switches. One bit tells only two things apart: yes and no, on and off. How many things can two bits tell apart?

Step 2 of 5

Place value

For two switches to hold a number, each place is given a value. In a decimal number like $37$⁠ the 3 is worth thirty, because it is in the tens place. In base 2 each place is worth twice the place to its right: the rightmost is worth $1$⁠, the next $2$⁠, then $4$⁠, $8$⁠ and so on.

On the board A is the 2s place and B the 1s place, so the switches are worth

$A \times 2 + B \times 1$⁠

Number in the controls is that value in base ten, and the switches show it in base 2. Predict, and then Number will move to $2$⁠.

Step 3 of 5

One one

To keep the bases apart, the base is written as a subscript: $10_2$⁠ is “one zero in base 2”, that is two, and a plain $10$⁠ is ten. Now go the other way: move Number until the switches show $11_2$⁠, and read what it is in base ten.

Step 4 of 5

More bits

Two bits only reach $3$⁠. To hold bigger numbers you add bits on the left, each worth twice the one before. Four bits are the places

$8 \quad 4 \quad 2 \quad 1$⁠

To read a binary number, write the place values over the bits and add the ones with a 1 under them. A 0 bit adds nothing, but it holds the place, just like the 0 in $305$⁠.

Step 5 of 5

The largest number

Each extra bit doubles the number of patterns, because every pattern of the earlier bits comes once with a 0 and once with a 1. So $n$⁠ bits give

$2^n$⁠

patterns, and they are the numbers $0$⁠ to $2^n - 1$⁠. Eight bits are a byte: $2^8 = 256$⁠ values. You will meet that number a lot.