The decimal system feels like the only "natural" way to count — but it's just one choice out of infinitely many. This guide explores how counting in base 2, 5, 7, and 8 works, where our own base-10 system came from, and how zero was invented.
Every number system you've ever used relies on a hidden choice: how many different digits are allowed before you have to carry over into a new column. That choice is called the base, and changing it changes everything about how numbers are written — while the quantities themselves never change at all.
A number base (also called a "radix") is the count of unique digits a system uses before it must move to a new column. Our everyday system uses ten digits (0–9), so it's base 10. Once you run out of single digits, you carry over — exactly the same idea works in every other base, just with a different digit limit.
In any base, each column represents a power of that base, starting from the rightmost column at power 0.
| Base | Name | Digits Used |
|---|---|---|
| 2 | Binary | 0, 1 |
| 5 | Quinary | 0, 1, 2, 3, 4 |
| 7 | Septenary | 0, 1, 2, 3, 4, 5, 6 |
| 8 | Octal | 0, 1, 2, 3, 4, 5, 6, 7 |
| 10 | Decimal | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
The base-10 positional system we use today — where a digit's column determines its value, and ten digits are enough to write any number — traces back to India. Early forms of these numerals, known as Brahmi numerals, were in use by around the 3rd century BCE, and evolved over the following centuries into what we now call Hindu-Arabic numerals.
The system's most likely reason for using base 10 in the first place is refreshingly simple: humans have ten fingers, and finger-counting is probably the oldest counting tool in human history.
These Indian numerals spread west through the work of Persian mathematician Muhammad ibn Musa al-Khwarizmi, working in Baghdad's House of Wisdom around 825 CE. His writings introduced Hindu numerals to the Islamic world — and his own name is the origin of the word "algorithm." From there, the Italian mathematician Fibonacci (Leonardo of Pisa) encountered the system while trading in North Africa and introduced it to Europe in his 1202 book Liber Abaci, gradually replacing Roman numerals across the continent.
Zero had to be invented twice, in two different ways — first as a placeholder, and much later as a true number.
The earliest use was as a placeholder: Babylonian mathematicians, around the 3rd century BCE, used a symbol to mark an empty column in their base-60 number system — signaling "nothing here" so numbers wouldn't be misread. The Maya civilization, entirely independently, developed their own zero symbol for their base-20 calendar system around a similar era. In both cases, zero was a placeholder, not yet a number you could add, subtract, or calculate with.
The decisive leap came from the Indian mathematician Brahmagupta, who in 628 CE wrote down formal rules for treating zero as a number in its own right, in his book Brahmasphutasiddhanta. He described what happens when zero is added to or subtracted from another number, and explored (with some errors later corrected by others) what happens when dividing by zero. This was the moment zero stopped being just a placeholder and became a full member of the number system.
An even earlier possible zero symbol appears in the Bakhshali manuscript, an ancient Indian mathematical text. Its exact age is debated by historians — radiocarbon dating of some fragments suggests portions may date as far back as the 3rd or 4th century CE, though scholars disagree on how to interpret the results.
The word "zero" itself traveled almost as far as the concept: the Arabic word "sifr" (meaning "empty") became the Italian "zefiro," which was eventually shortened to "zero" — the same Arabic root also gives English the word "cipher."
Binary uses only two digits, 0 and 1. It's the language computers speak internally, because electronic circuits are simplest when they only need to represent two states: off (0) and on (1).
The binary number 1101 means: (1×8) + (1×4) + (0×2) + (1×1) = 8+4+0+1 = 13 in decimal.
Base 5 uses five digits: 0, 1, 2, 3, 4. It's a natural fit for counting on the fingers of just one hand, and some human counting systems throughout history are believed to have used it for exactly that reason.
The base-5 number 132 means: (1×25) + (3×5) + (2×1) = 25+15+2 = 42 in decimal.
Base 7 uses seven digits: 0 through 6. It's rarely used as a full counting system historically, but it's an excellent base for practicing the general pattern of base conversion, since it doesn't line up conveniently with powers of 2 or 10 the way other bases do.
The base-7 number 245 means: (2×49) + (4×7) + (5×1) = 98+28+5 = 131 in decimal.
Octal uses eight digits: 0 through 7. It's historically important in computing, since 3 binary digits map perfectly onto 1 octal digit — making octal a convenient shorthand for reading long strings of binary. It's still used today in Unix and Linux file permission codes.
The octal number 157 means: (1×64) + (5×8) + (7×1) = 64+40+7 = 111 in decimal.
It's easy to forget that each base restricts which digits are even allowed.
The fix: octal only allows digits 0–7. The digit 8 doesn't exist in base 8 — the same way base 10 has no single digit for "ten." "381" is not a valid octal number because of the 8.
Decimal, the system used for everyday counting, uses ten digits: 0 through 9. Every number system in this guide works by exactly the same logic as decimal — only the digit limit and the powers involved actually change.
The decimal number 4,502 means: (4×1000) + (5×100) + (0×10) + (2×1) = 4,502. This is the same expansion method used for every other base — just with powers of 10 instead of powers of 2, 5, 7, or 8.
Because binary digits look like decimal digits, it's tempting to read them the same way — but the place values underneath are completely different.
The fix: binary 101 = (1×4) + (0×2) + (1×1) = 5 in decimal, not one hundred and one. Always expand using the base's own powers, never assume decimal place values apply.
To convert a number from any base into decimal, multiply each digit by its column's power of the base, then add everything together. This is the same method used in every worked example above — it always works, for any base.
| Number | Base | Expansion | Decimal Value |
|---|---|---|---|
| 1101 | 2 | (1×8)+(1×4)+(0×2)+(1×1) | 13 |
| 132 | 5 | (1×25)+(3×5)+(2×1) | 42 |
| 245 | 7 | (2×49)+(4×7)+(5×1) | 131 |
| 157 | 8 | (1×64)+(5×8)+(7×1) | 111 |
To go the other way — converting a decimal number into another base — divide repeatedly by the target base, writing down each remainder. Once the answer to division reaches zero, read the remainders from the bottom to the top. That order matters enormously.
25 ÷ 2 = 12 remainder 1
12 ÷ 2 = 6 remainder 0
6 ÷ 2 = 3 remainder 0
3 ÷ 2 = 1 remainder 1
1 ÷ 2 = 0 remainder 1
Reading the remainders bottom-to-top: 11001
This is the single most common mistake in the repeated division method — the remainders come out in reverse order.
The fix: the very last remainder you calculate is actually the first digit of your answer. Always read from the bottom of your division steps upward. Reading top-to-bottom for 25 ÷ 2 would incorrectly give 10011 instead of the correct 11001.
Tap any decimal number below to see it instantly converted into binary, base 5, base 7, and octal.
Every photo, song, and app on a computer is ultimately stored as binary — long strings of 0s and 1s. Because raw binary is hard for humans to read, programmers often use octal or hexadecimal as a shorthand. Unix and Linux systems still use octal today for file permission codes, like "chmod 755," where each digit controls read, write, and execute access.
The German mathematician Gottfried Leibniz developed and refined binary arithmetic in the late 1600s, describing it in a 1703 paper — over 200 years before it became the foundation of modern computing.
Not every historical culture used base 10. The ancient Babylonians used base 60 (which is why a circle has 360 degrees and an hour has 60 minutes), and the Maya used base 20 (possibly counting on both fingers and toes).
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