Names of Large Numbers: Notation, Scales, and Uses

Scientific and engineering notation

Scientific notation writes a number as a mantissa between 1 and 10 multiplied by 10 raised to an integer exponent. For example, 5×10^20 denotes 500,000,000,000,000,000,000 and 6.425×10^11 denotes 642,500,000,000. This form highlights the order of magnitude and makes multiplication, division and power operations clearer. A related convention, engineering notation, restricts the exponent to multiples of three so that the exponent aligns with common grouping names and SI prefixes.

Grouping and word names (scales)

Grouped-word naming assigns a name to each group of three digits. Two historically important scales exist: the short scale used in many English-speaking countries and the long scale traditionally used in parts of continental Europe. In the short scale, each new term greater than million indicates a thousandfold increase of the previous term: billion = 10^9, trillion = 10^12, quadrillion = 10^15, quintillion = 10^18, and so on. In the long scale, used in some languages and historically in Europe, a billion traditionally meant 10^12 while 10^9 was called a milliard (or sometimes 'thousand million'). Because of these differences, the same word can mean different powers of ten in different regions; see example links for common regional usages: American short scale examples and European long scale examples.

Examples and common conversions

• 500,000,000,000,000,000,000 = 5×10^20 = five hundred quintillion (short scale).
• 642,500,000,000 = 6.425×10^11 = six hundred forty-two billion, five hundred million (short scale); in some long-scale usages this might be spoken as six hundred forty-two milliard, five hundred million.
• 10^3 = thousand; 10^6 = million; 10^9 = billion (short scale) or milliard (long scale); beyond these, names proceed by groups of three digits.

For clarity when communicating across regions, writers often combine a numeric form with a word form (for example, "6.425×10^11, about 642.5 billion") or use explicit powers of ten.

SI prefixes and very large orders

Standards bodies provide short prefixes for powers of ten in steps of three, widely used in science and engineering: kilo- (10^3), mega- (10^6), giga- (10^9), tera- (10^12), peta- (10^15), exa- (10^18), zetta- (10^21) and yotta- (10^24). Larger prefixes have been introduced more recently for extremely large quantities; these prefixes are convenient alternatives to long word names in technical contexts.

Practical notes, history and conventions

The practice of naming large numbers has evolved over centuries. Early naming systems arose from counting needs in commerce and astronomy; later, mathematicians and standardizing organizations refined names and introduced decimal and metric conventions. In practice, scientists prefer powers-of-ten notation and SI prefixes for precision, while journalists and the public often use grouped names. When precision and international clarity matter, it is good practice to include the numeric form (explicit digits or exponent) alongside any worded name to avoid ambiguity.