Forty-two things stay forty-two things. A base changes how we group and write that quantity. Learn the pattern behind place value, then follow a conversion one step at a time.
Commas separate place digits in bases 20 and 60 on this page.
In a positional number system, the base tells you how many units make one unit of the next place. In base 10, ten ones make a ten, and ten tens make a hundred. In base 2, two ones make a two, and two twos make a four.
The rightmost whole-number place is always worth 1. Each step left multiplies that value by the base: 1, base, base², base³, and so on. A digit tells you how many of that place you have. Zero holds an empty place.
| Place value | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|
| Digit | 1 | 0 | 1 | 0 | 1 | 0 |
| Contribution | 32 | 0 | 8 | 0 | 2 | 0 |
32 + 8 + 2 = 42
Each digit must be smaller than its base. Binary has 0 and 1; after 1 comes 10, meaning one two and zero ones. So 10 does not always mean ten.
NASA · Place value in binary ↗First read the source digits using their place values. Then divide by the target base to find its digits. These are general conversion methods; individual civilizations had their own arithmetic practices.
For bases 20 and 60: type each place digit as a decimal number, separated by commas. For example, 1,12 in base 60 means 1 × 60 + 12 = 72. The comma is a place separator, never a thousands separator here. Spaces around commas are allowed. Fractions, signs, prefixes, colons, and semicolons are outside this lesson.
4210 1010102
Decimal value: 42. The small subscript names the base; it is not part of the number.
Start with 1 at the right. Multiply each place value by 10 as you move left, then add each digit’s contribution.
| Digit | Place value | Contribution |
|---|---|---|
| 4 | 101 = 10 | 4 × 10 = 40 |
| 2 | 100 = 1 | 2 × 1 = 2 |
40 + 2 = 42
Divide the decimal value by 2. Keep the remainder, divide the quotient again, and stop when the quotient is 0.
| Divide by 2 | Quotient | Remainder |
|---|---|---|
| 42 ÷ 2 | 21 | 0 |
| 21 ÷ 2 | 10 | 1 |
| 10 ÷ 2 | 5 | 0 |
| 5 ÷ 2 | 2 | 1 |
| 2 ÷ 2 | 1 | 0 |
| 1 ÷ 2 | 0 | 1 |
↑ Read the remainders from bottom to top.
1010102
42 in base 10 equals 101010 in base 2
Number systems grow from counting habits, language, measurement, and the tools people use. A useful convention can be passed down for centuries. We often know how a system worked more confidently than why its earliest users settled on a particular grouping.
Ten fingers provide a convenient counting aid, a plausible reason for the widespread use of tens. That is not proof of one shared origin for every decimal tradition. Egypt’s grouping by tens and modern decimal place value use different writing rules.
MacTutor · The question of number bases ↗Counting fingers and toes gives twenty, a common explanation for vigesimal grouping. Treat it as an explanation, rather than a proven origin story for the Maya. Maya numerals use dots, bars, and zero signs and supported trade, astronomy, and calendar work.
Here base 20 means pure powers of twenty. The Maya page explains the distinction from calendar place values.
Smithsonian · Maya mathematics ↗Babylonian scribes inherited sexagesimal traditions from the Sumerians. The original reason for sixty remains uncertain. Its many divisors do make sharing convenient: a half, a third, and a quarter are 30, 20, and 15 sixtieths. Usefulness is not proof of origin.
Familiar traces remain in sixty minutes to an hour and sixty seconds to a minute. Historical cuneiform readings also depended on context to establish scale.
MacTutor · Babylonian numerals ↗Digital circuits can represent two distinguishable states, such as low and high voltage, as 0 and 1. Designing around two separated states makes reliable switching practical. Each binary digit is a bit. Several bits together can represent numbers, letters, colors, instructions, and other information.
IBM · Bits and their physical representation ↗Long binary strings are hard to read. Since 16 = 2⁴, exactly four bits fit into one hexadecimal digit. Hex uses 0–9, then A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. After F comes 10, meaning one sixteen and zero ones.
Hex is a compact way for people to read binary data, used in places such as color codes and memory addresses. It does not require a different kind of sixteen-state computer hardware.
NASA · Why hexadecimal fits binary ↗| Decimal | Hex | Binary |
|---|---|---|
| 0 | 0 | 0000 |
| 1 | 1 | 0001 |
| 2 | 2 | 0010 |
| 3 | 3 | 0011 |
| 4 | 4 | 0100 |
| 5 | 5 | 0101 |
| 6 | 6 | 0110 |
| 7 | 7 | 0111 |
| 8 | 8 | 1000 |
| 9 | 9 | 1001 |
| 10 | A | 1010 |
| 11 | B | 1011 |
| 12 | C | 1100 |
| 13 | D | 1101 |
| 14 | E | 1110 |
| 15 | F | 1111 |
Group binary digits in fours from the right. Add leading zeros to the leftmost group if needed; they do not change the value. Match each group to one hex digit. To go back, replace each hex digit with its four bits. Eight bits make one byte.