1.1 Ten symbols

We pick a set of symbols with an arbitrary integral meaning for each.

For example, the symbol 3 means three things; if we’re counting 🧡s, 3 means 🧡🧡🧡.

These meanings need to be consecutive, and we need one that means zero.

The specific symbols we use are:

There are ten symbols in this list, which is why we call them base-ten numbers.

1.2 In order

We arrange several of these symbols in a specific order. One end of the ordering is the big end and the other is the little end.

The specific order we use puts the little end on the right and the big end on the left²² Incidentally, the standard of little end on the right applies in both left-to-right languages like English and right-to-left languages like Arabic. From a global, multilingual perspective it’s big on left, not big first..

1.3 With place-dependent meaning

We change the meaning of the symbols based on where they are in the ordering: that is, their value depends on their place in the list of symbols, which is why we call them place-value numbers.

For example, in the number 121 the first symbol 1 does not mean the same thing as the last symbol 1, even though they are the same symbol.

The little end symbol has the meaning we gave above.

Each other symbol’s value is ten times the value it would have if it were one place closer to the little end: ten, again, because these are base-ten numbers.

Thus, to figure out what 121 🧡s means:

• The low-end symbol is 1, which means 🧡

• The next symbol is 2, which normally means 🧡🧡 but in this place means ten times that: 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡

• The next symbol is 1, which normally means 🧡 but in this place means ten times ten times that: 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡

• Adding those together gets the full meaning: 🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡 🧡🧡🧡🧡🧡🧡🧡🧡🧡🧡

1.4 Combined in mathematical notation

The meaning of symbol $s$ in a position $k$ symbols to the left of the right end of the number is $s \times \text{ten}^k$.

Sometimes this is conveniently written out in tabular form, so for the Fall 2025 total student enrollment at the University of Illinois Urbana-Champaign of 60848 we might write:

1.5 Grouping symbols

While place value numbers are very useful, far superior to early number systems from the perspective of doing arithmetic, they can get unwieldy if they get too long. For example, for a fraction of a second near the beginning of 2026 the US national debt was 38558175645385. That number has so many digits we get lost in the middle and have trouble even telling its order of magnitude.

A tool for helping people understand such numbers is to group digits in some predictable spacing; in English it’s common to group by 3 digits, giving us the more-understandable 38,558,175,645,385 instead.

Grouped digits can also be considered as if the groupings were themselves individual symbols. There are exactly one thousand possible groupings of three base-ten digits, which are adjacent and include a zero (000), which means we can use the same place-value logic on the resulting base-one-thousand numbers as we did on base-ten numbers. For example, in this form the debt becomes

2.1 Easiest base for hardware: base two

The most fundamental operation digital computers engage in is arithmetic on numbers. How they do this is beyond the scope of this course, but we can get the right intuition with a hand-wavy discussion.

While the exact numbers in the example above are not precise, the overall pattern holds: smaller bases lead to smaller, more efficient, less expensive machines, with base 2 being the most efficient base of all.³³ Base two is also more efficient for humans: no need to learn big multiplication tables or try several guesses per digit during long division: each step is simple and straightforward with no more than 4 (and usually just 2) choices at each step. That said, you should still teach your children base ten, not because it’s better but because it’s common. It’s more useful to know what other people know than to know a better tool that you can’t use to communicate with most people.

Let’s work through base-two place-value numbers:

We need two symbols

Places increase by a factor of two instead of ten as they move to the left

Thus 1100110 in base two means the same thing as 102 in base ten: $$\begin{array}{rl} 1100110\text{ in base two}=&1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\\ =& 2^6 + 2^5 + 2^2 + 2^1\\ =& 64 + 32 + 4 + 2\\ =& 102 \text{ in base 10} \end{array}$$

The implementation efficiency of base two is the origin of the statement in a computer, everything is 0s and 1s. It’s a similar kind of statement to on a book, everything is white paper and black ink: it’s not false, exactly, but also not super insightful.

The word binary means related to two and is used in computing both as a shorthand label for base-two place-value numbers, and for data stored in a computer-friendly format, and for data that makes sense to a computer program but not to most humans. Binary is also used outside of computing to modify other words to emphasize two, such as a binary decision (a choice between one of two things with no middle-ground).

A single base-two digit is called a bit, a portmantau of binary digit.

2.2 Digit grouping for bigger data: base 256

One bit of data doesn’t tell us very much; almost any interesting problem a computer might be asked to do needs multi-bit inputs and outputs. Early in computing it was decided to group bits together for most purposes (other than actually doing computation) into large enough groups to be useful.

And what did useful mean?

The primarily British and US engineers working on early computers decided to have one digit be big enough to represent one action of a mechanical typewriter: type a letter, number, punctuation mark, advance or back up the print head, and so on. They found about 100 things to do, rounded up to have room for later additions, and chose 256, which is a grouping of 8 bits. They named each such 8-bit grouping a byte; thus a byte is eight bits, a number between 0 and 255, a digit in base-256 place-value numbers, or something else (such as an action on an English typewriter) represented by such a number.⁴⁴ In some technical documents, the word octet is used instead of byte because in early computing history not all hardware used 8-bit bytes so octet (from the Greek word for eight) is seen as more precise.

Bytes aren’t just a convenient way of writing: it’s built into almot all hardware. The inner workings of one piece of modern computers (the processor) works on the bits of base-2 numbers; everything else (memory, disks, displays, networks, etc) works on the bytes of base-256 numbers.

2.3 Easier to read: base 16

Bits is useful for computation, but it’s easy to get lost reading binary numbers; even smallish numbers like 58 are long enough binary numbers (111010) that it’s easy to mistake them for other numbers (such as 1111010 = 122).

Bytes are useful for data storage, but 256 is too many values to try to learn a symbol for each one and 8 is too many bits for separators like commas to be readily readable.

The compromise used by computing professionals is base 16. From a bits perspective, that’s 4 bits, so base 16 is just a digit grouping of base 2. From a bytes perspective, that’s half a byte, so base 256 is just a digit grouping of base 16. And learning 16 symbols is much more doable.

We call base-sixteen hexadecimal or simply hex.

The hexadecimal symbols that are used are the ten base-ten digits followed by the English letters A through F (in either upper or lower case).

Hex numbers make it into end user experiences in some unexpected places, especially when the computer is exposing something like byes, such as internal color codes or a computer’s unique identifier on a network.