Describe digital logic levels, explain binary representation, and convert between small binary and denary numbers

What this dot point is asking

SEAB wants you to describe the two digital logic levels, to explain how binary represents information, and to convert small numbers between binary and denary (everyday base ten). The central insight is that digital electronics uses only two voltage levels, called logic 0 and logic 1, because two well-separated levels are reliable and easy to process, and that any number or instruction can be coded as a pattern of these two states.

The answer

The two logic levels

A digital signal is allowed only two voltage levels:

• Logic 0 is a low voltage, near $0\ \text{V}$.
• Logic 1 is a high voltage, near the supply voltage.

There is a forbidden gap in between that the signal passes through quickly but never rests in. These two states are also called low and high, or off and on. Everything in a digital circuit is built from patterns of 0s and 1s.

Why two levels

Using only two levels makes digital circuits robust. Electrical noise adds small unwanted voltages, but because logic 0 and logic 1 are far apart, a little noise does not turn a clear 0 into a 1 or the other way round. Two levels are also easy to store, copy and process without the signal degrading, which is why computers, phones and memory all work digitally.

Binary numbers

Binary is the base-two number system, using only the digits 0 and 1, which matches the two logic levels exactly. Each binary digit (bit) has a place value that is a power of two. For a four-bit number the place values, from left to right, are:

So the binary number $1101$ means $(1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) = 13$.

Converting between binary and denary

• Binary to denary: write the place values above the bits, multiply each bit by its place value, and add them up.
• Denary to binary: find the largest place value that fits, put a 1 there, subtract it, and repeat with the remainder, putting 0s where a place value does not fit. For example, $10$ in denary is $8 + 2$, so the bits at the $8$ and $2$ places are 1 and the rest are 0, giving $1010$.

Examples in context

Example 1. A four-bit counter display. A digital counter holds a number as four bits, lighting LEDs to show the pattern. The bits $0110$ light the LEDs at the $4$ and $2$ places, which a reader decodes as $6$. The two-level system means each LED is simply on or off, with no in-between brightness to misread.

Example 2. Reliable data over a long cable. Sending a digital signal down a long, noisy cable works because the receiver only has to decide whether each level is high or low, not measure an exact voltage. Noise that would ruin an analogue signal leaves the digital 0s and 1s readable, which is why digital communication dominates modern electronics.

Try this

• Cue. Convert $0101$ to denary. Place values $8, 4, 2, 1$ give $(0 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) = 5$.

• Cue. Convert the denary number $12$ to four-bit binary. $12 = 8 + 4$, so the $8$ and $4$ places are 1: $1100$.

• Cue. State the two logic levels and roughly what voltage each represents. Logic 0 is a low voltage near $0\ \text{V}$; logic 1 is a high voltage near the supply voltage.