Binary and Base Systems: The On-Off Math of Computers

The letter A on a calculator that had no business showing one

My daughter borrowed my old calculator, found the switch marked HEX, and turned it. The display that had been showing 4 and 7 all morning started showing letters — A, then C, then F. Same buttons, same screen. She asked why a calculator needs a whole extra alphabet to say a number, and I gave the parent answer: because that is how they made it.

It was not arbitrary, and it was not about counting to ten. The answer to her question is the same reason every computer, phone and digital clock works this way: base ten is a habit, not a law, and base two is the easiest base to build in.

A base is the size of your box, nothing more

Start with what you already know. Write a number and the position of each digit decides what it is worth. In the 3 of 34, the 3 is worth three tens and the 4 is worth four ones. That column system is place value, and the multiplier between columns is all that is at issue.

In base ten that multiplier is ten. Change it to five and every column becomes five times the one to its right. Change it to two and each column is double. Nothing else changes. The boxes just got smaller. So the real question is not which base is right. It is: how many symbols do you want on one column?

That count is the base. Pick twelve and you need twelve symbols, so 9, 10 and 11 take a second character — which is where the letters in older counting systems came from. Pick sixty and an hour splits into sixty minutes, as it does today.

Ten is a hand, not a theorem

The usual explanation for base ten is fingers, and that is almost certainly where it came from.

Base twelve wins wherever things get counted by the dozen — eggs, flowers, bottles. It divides evenly by 2, 3, 4 and 6, which ten does not. That is why we keep 12 hours on a clock and 24 on a calendar.

Base sixty survives for the same reason. Sixty divides by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, so it breaks cleanly into halves, thirds, quarters and fifths. A minute has 60 seconds and a circle 360 degrees.

Both are more convenient than ten. We kept ten anyway, because the cost of changing is enormous and the benefit is invisible. Bases are not chosen because they are best. They are kept.

The same quantity, different string

Here is the part that makes people uneasy. Write thirty-five in base ten and you get 35. Write it in base two and you get 100011, which looks nothing like thirty-five. Nothing was added or lost — only the packaging changed.

The base-ten version hides this, because 3 sits in the tens column and 5 in the ones column. In base two the columns are only ones and twos, so you spell out each power of two separately: 32 + 2 + 1 = 35. Add those and you land on the same amount. That is all base two is — place value whose columns happen to be powers of two, the numbers a machine holds most easily. Our place value guide covers the base-ten version of the same idea.

Go the other way and the method is mechanical. Divide by two, write down the remainder, divide again, read the remainders upward. Standard school division pointed at a different divisor, the routine our long division guide builds with ten.

Why two, when two is the worst choice for people

Base two needs no arithmetic at all to build. A switch is on or off. A wire carries voltage or it does not. One physical object, two distinguishable states — the cheapest reliable thing you can manufacture, and you can make billions of them.

A machine holding ten states per wire needs hardware that can tell ten apart: bigger, harder to calibrate, wrong more often. Base two is not chosen because two is elegant, but because it is the easiest state to distinguish, and a bit that cannot be read reliably is worth nothing.

One nuance worth having: the voltage on a real wire is not exactly zero or exactly full. A range counts as off, another as on. And when the CPU needs more than one bit it groups them — eight make a byte, 0 to 255, written base two, read as base ten.

Binary arithmetic is genuinely bad

Here is the part that surprises adults. In base two, 2 + 2 = 100. Adding one to one gives ten, because one plus one fills a box and rolls over. Every second sum carries. Counting to ten takes four symbols. Fractions get uglier still: a fraction has a finite binary form only when its denominator, in lowest terms, is a power of two. One half is easy. One third never ends. Our fraction guide stays in base ten, where thirds are respectable.

So a computer built on binary is bad at arithmetic — not a little, but structurally. Not a flaw, a consequence of choosing two. When your calculator shows 0.1 + 0.2 = 0.30000000000000004, you are watching a machine that counts only in twos meet a world that insists on tenths.

The fix is dedicated hardware that handles decimal directly. Every phone and laptop has one, and we never think about it because it works. It is also why you will hear people insist computers “really do” use binary for maths. They use binary, then add a translator.

Sixteen is four bits in a trench coat

Humans find long strings of ones and zeroes hard to read, and a mistyped digit is invisible. So people almost never write binary directly — they use base sixteen, where every hex digit is exactly four binary digits. 1111 becomes F, 10000 becomes 10, and that is the whole trick. No new mathematics: a wider box for the same machine, since 16 = 2 to the power of 4, so columns line up with the bytes underneath.

So the calculator was right and my answer was wrong. Base sixteen became a display standard for a hardware reason: a seven-segment display lights only seven bars, and A through F were added precisely because they could be built from those same bars. Hex won on screen the way it won on paper — it fits the machine it has to talk to. Stylesheets do the same: #FF5733 is three pairs of hex digits, one byte each.

Try it on these three

First, write thirteen in base two: divide by two three times, read the remainders up — 1101. Then count that thirteen in base ten by doubling as you go. Second, look at your kitchen clock and work out why the minute hand moves in sixties. Third, count to ten in base two on your fingers. You run out at three, which is not a puzzle failure — it is the clearest proof that base ten suits hands, not minds.

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