Base Camp

Not a third system, a shorter notation

Most people meet hexadecimal and file it away as a third number system to learn alongside binary and decimal. ⚠️ It is not, and treating it as one makes this topic about three times harder than it needs to be. Here is the whole thing in two lines. Four bits have exactly sixteen possible combinations. Hexadecimal has exactly sixteen digits. So one hex digit is four bits. Not similar to four bits, not a way of approximating four bits. It is a single symbol standing in for a group of four, in the same way that a shorthand mark stands in for a whole word. ⚠️ And that gives you the most useful fact on this strand straight away: converting between binary and hex needs no arithmetic whatsoever. You are not calculating, you are transcribing. Decimal is the awkward one, because ten is not a power of two and so nothing lines up neatly.

Words for the three bases

Five terms. The last one is the reason the whole strand hangs together.

Which conversions need arithmetic

Three pairs of bases, three completely different amounts of work. Knowing which is which saves time and mistakes in an exam.

How to show working on a conversion

Your specification says it directly: always show your working on a conversion. That is not a politeness. It is where the marks are. ⚠️ A bare answer that is wrong scores nothing. Correct working with one slip in the final addition usually scores most of the marks. So the student who writes only a number is risking everything on getting it exactly right, for no gain at all. Write the place values down FIRST, before you look at anything else. Then mark which ones are used. Then add. Doing it in that order means the structure of the answer is on the page before you start making arithmetic mistakes in it. ⚠️ And sanity-check the size before you write the answer down. If the value you are converting can only reach a few hundred and yours has come out in the thousands, something has gone wrong, and you can catch that in about five seconds. Checking the magnitude is the cheapest mark-saver on this whole topic.

Match each four-bit group to its digit

Why a byte is two digits

Exactly one byte always fits into exactly two hexadecimal digits, with nothing left over and nothing to spare. Why?

• A byte is eight bits, and each hex digit stands for exactly four bits, so two digits cover all eight
• Because hexadecimal counts in sixteens and a byte holds sixteen different values
• Because two is the smallest number of digits that can hold a reasonably large number
• It is a convention rather than a rule, and other ways of splitting a byte would work just as well

One byte, three ways

Take the byte 00101111 and write it in all three bases, showing the working each time. To denary. Write the place values above the bits: 128, 64, 32, 16, 8, 4, 2, 1. Now see which ones have a 1 underneath: 32, 8, 4, 2 and 1. Add them: 32 + 8 + 4 + 2 + 1 = 47. To hexadecimal. Split the byte into two halves: 0010 and 1111. The first is 2. The second is fifteen, which hex writes as F. So the byte is 2F. ⚠️ Notice what just happened, and it is the point of this module. The denary conversion needed five numbers added together. The hex conversion needed no arithmetic at all - just a split and two lookups. So 00101111, 47 and 2F are the same value written three ways. ⚠️ And on notation: hex is written plainly here, as 2F. You will meet prefixes and subscripts elsewhere, so if a question does not make its notation clear, say which one you are using.

Add the place values

A byte holds 00011010. Using the place values 128, 64, 32, 16, 8, 4, 2 and 1, add together the ones that have a 1 beneath them to convert it to denary.

Order the conversion

Put the steps of converting a byte to denary into the order you should do them.

• Write the place values above the bits, largest on the left
• Mark which place values have a 1 underneath them
• Add those place values together
• Check the total is a sensible size for eight bits
• Write the answer down, leaving the working visible

Complete the number bases paragraph

The system using only the digits 0 and 1 is called _____. The everyday system built on ten digits is called _____. The system with sixteen digits, in which the letters A to F stand for the values ten to fifteen, is called _____. A group of four bits is called a _____, and that is exactly how much one hexadecimal digit represents.

Convert at speed

Five in a row, three lives. All of these follow from one hex digit being four bits.

Spot the true number base facts

Tap the TWO statements that are true.

• One hexadecimal digit stands for exactly four bits
• Converting binary to hexadecimal needs no addition
• Hexadecimal can represent values that binary cannot
• The letters in hexadecimal stand for values above sixteen

Three conversion calls

Three situations in an exam. Choose the answer you could defend.

• A student converts a byte to denary and writes down only the final number, which turns out to be wrong. What did they lose that they need not have lost?
• A student needs a byte written in hexadecimal, and begins by converting it to denary first. What is wrong with that approach?
• A student writes that hexadecimal exists because computers work in sixteens. What is the better explanation?

Explain the three bases and how to move between them

A friend has decided hexadecimal is a whole third number system to memorise and is dreading it. Write them the answer that makes it smaller than they think.

• Name the three number bases this topic uses and say how many digits each one has
• Explain why hexadecimal needs the letters A to F at all
• Explain why one hexadecimal digit corresponds to exactly four bits
• Show how to convert one byte into denary, setting the working out properly
• Finish by saying which conversion needs no arithmetic, and why that follows from what you have explained