Decimal: 347
3 × 10² + 4 × 10¹ + 7 × 10⁰
Three hundreds, four tens, and seven ones equal 347.
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Number Systems
Compare the bases, digit sets, place-value rules, and historical notations people use to represent numbers. Start with the quick chart, then open a focused Number Digit guide for symbols, examples, and tables.
Quick Answer
A number system is a structured way to represent numbers. A positional system uses a base and gives each digit a value based on its position. Decimal is base 10, binary is base 2, octal is base 8, hexadecimal is base 16, and vigesimal is base 20.
Not every notation works this way. Roman numerals combine fixed-value symbols, while Arabic and Devanagari digits are different symbol sets that can both be used to write decimal numbers.
Number System Comparison Chart
This chart separates a system's base from the symbols used to display it. In a positional system, the base tells you how many single-digit values are available before a new place begins.
| System | Base | Digits | Decimal 42 | Common context |
|---|---|---|---|---|
| Binary | 2 | 0–1 | 101010 | Digital logic and bit-level data |
| Octal | 8 | 0–7 | 52 | Compact three-bit groups and Unix permissions |
| Decimal | 10 | 0–9 | 42 | Everyday counting and measurement |
| Hexadecimal | 16 | 0–9, A–F | 2A | Programming, addresses, and color values |
| Vigesimal | 20 | Twenty digit values | 2220 | Mayan and Kaktovik numeral study |
A subscript can identify the base when a written value could be ambiguous. For example, 2220 means two twenties plus two ones, which equals 42 in decimal.
How Base and Place Value Work
In positional notation, a digit does not carry its full value alone. Its column multiplies it by a power of the base. Reading from right to left, the columns represent base0, base1, base2, and so on.
Binary: 1011
1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰
Eight plus zero plus two plus one equals decimal 11.
Hexadecimal: 2A
2 × 16¹ + 10 × 16⁰
Two sixteens plus ten ones equals decimal 42.
One Number, Different Numerals
A number is an abstract value; a numeral is the written expression of that value. The quantity twelve stays the same in every row below, even though its notation changes.
| Notation | Written form | What changed? |
|---|---|---|
| Decimal with Western digits | 12 | Base 10 positional notation |
| Binary | 1100 | The base changed to 2 |
| Hexadecimal | C | The base changed to 16 |
| Roman numerals | XII | A non-positional symbol system |
| Eastern Arabic digits | ١٢ | The decimal digit shapes changed |
| Devanagari digits | १२ | The decimal digit shapes changed |
Types of Numeral Systems
Positional systems
A symbol's position changes its value. Decimal, binary, octal, hexadecimal, and most modern base systems belong here. Zero often serves both as a number and as a placeholder.
Additive or subtractive systems
Symbols have fixed values and are combined according to ordering rules. Standard Roman numerals are a familiar example: VIII adds values, while IX subtracts one from ten.
Digit scripts
Different writing systems may supply distinct glyphs for the same decimal digit values. Western 0–9, Eastern Arabic ٠–٩, Persian ۰–۹, and Devanagari ०–९ can all express base-10 place value.
Grouping and naming systems
The Indian and international systems both use decimal notation, but they group and name large values differently. Compare 10,000,000 with 1,00,00,000, or ten million with one crore.
A Practical Guide to Reading Number Systems
When you meet an unfamiliar numeral, begin by asking three separate questions: what value is being represented, what base controls the place values, and which symbols are being used? Keeping those questions apart prevents a common mistake. A new-looking set of digits does not automatically mean a new base. The Devanagari numeral २५ and the Western-digit numeral 25 use different glyphs, yet both can represent two tens and five ones in decimal notation.
The base matters most in a positional system. Base 10 has ten digit values, so each column is ten times the column to its right. Base 2 has only zero and one, and each column is twice the next. Base 16 extends the available single-digit values with A through F. This is why the same visible sequence may mean different quantities when its base is not stated: 101 is one hundred one in decimal, five in binary, and seventeen in hexadecimal.
Why several bases remain useful
Decimal fits everyday human calculation because it is familiar and works smoothly with common measurement and money formats. Binary matches two-state digital hardware. Octal compresses each group of three binary digits into one symbol, while hexadecimal compresses four binary digits into one symbol. Base 20 is valuable historically and culturally, and it also shows that place-value reasoning is not limited to powers of ten.
Historical numeral systems reveal other solutions. Roman notation assigns fixed values to letters and builds a numeral by adding or subtracting selected combinations. It is effective for labels, dates, outlines, and inscriptions, but it lacks the ordinary zero placeholder and column structure of modern decimal notation. Mayan numerals use base-20 ideas with their own visual arrangement. These systems are not failed versions of decimal; they are distinct designs shaped by different needs and traditions.
Symbols, language, and formatting
Written numbers also depend on script and regional convention. Eastern Arabic, Persian, and Devanagari digits have their own Unicode characters, which matters when copying, searching, programming, or designing fonts. Meanwhile, the Indian number system changes comma grouping after the hundreds place and uses names such as lakh and crore. The underlying arithmetic can remain decimal even when the symbols, separators, and spoken scale names differ.
For reliable interpretation, look for a base label, examine the allowed digits, identify whether position affects value, and check the cultural or technical context. A value beginning with 0x is commonly hexadecimal in programming; a string containing only zeros and ones may be binary, but context is still required; and a sequence such as XII follows Roman rules rather than decimal columns. Use the comparison chart above for orientation, then open the focused guides for complete symbol tables and worked examples.
Number Systems FAQ
What is a number system?
A number system is an organized way to represent numbers. It may define a base, a set of digits or symbols, place-value rules, and conventions for combining those symbols.
What are the four common positional number systems?
The four systems most often compared in computing are binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16).
What is the difference between a number system and a numeral system?
The terms often overlap. In careful use, a number system can also mean a mathematical collection such as the integers or real numbers, while a numeral system is specifically a notation for writing numbers.
Are Arabic and Devanagari digits different number bases?
No. Arabic and Devanagari digit sets can both write ordinary base-10 positional numbers. The visible symbols change, but the place-value structure can remain decimal.
Is the Indian number system a different base?
Modern Indian notation is decimal. Its distinctive feature is grouping and naming large place values, such as lakh and crore, rather than changing the base from 10.
Why does hexadecimal use letters?
Base 16 needs sixteen single-digit values. After 0 through 9, the letters A through F conventionally represent decimal values 10 through 15.
Do all numeral systems use place value and zero?
No. Modern decimal and binary are positional and use zero as a digit. Standard Roman numerals are non-positional and have no standard zero symbol.
What number system do computers use?
Digital computers represent low-level states with binary. Octal and hexadecimal are compact ways for people to read groups of binary digits, while decimal remains common in user-facing values.