Exponentiation in the context of "Archimedes"

Trillion is a number with two distinct definitions:

• 1,000,000,000,000, i.e. one million million, or 10 (ten to the twelfth power), as defined on the short scale. This is now the meaning in both American and British English.
• 1,000,000,000,000,000,000, i.e. 10 (ten to the eighteenth power), as defined on the long scale. This is one million times larger than the short scale trillion. This is the historical meaning in English and the current use in many non-English-speaking countries where trillion and billion 10 (ten to the twelfth power) maintain their long scale definitions.

A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers: 1, 2, 3, 4, 5, and so forth. Individual numbers can be represented in language with number words or by dedicated symbols called numerals; for example, "five" is a number word and "5" is the corresponding numeral. As only a limited list of symbols can be memorized, a numeral system is used to represent any number in an organized way. The most common representation is the Hindu–Arabic numeral system, which can display any non-negative integer using a combination of ten symbols, called numerical digits. Numerals can be used for counting (as with cardinal number of a collection or set), labels (as with telephone numbers), for ordering (as with serial numbers), and for codes (as with ISBNs). In common usage, a numeral is not clearly distinguished from the number that it represents.

In mathematics, the notion of number has been extended over the centuries to include zero (0), negative numbers, rational numbers such as one half ${\displaystyle \left({\tfrac {1}{2}}\right)}$, real numbers such as the square root of 2 ${\displaystyle \left({\sqrt {2}}\right)}$, and π, and complex numbers which extend the real numbers with a square root of −1, and its combinations with real numbers by adding or subtracting its multiples. Calculations with numbers are done with arithmetical operations, the most familiar being addition, subtraction, multiplication, division, and exponentiation. Their study or usage is called arithmetic, a term which may also refer to number theory, the study of the properties of numbers.

Large numbers are numbers far larger than those encountered in everyday life, such as simple counting or financial transactions. These quantities appear prominently in mathematics, cosmology, cryptography, and statistical mechanics. Googology studies the naming conventions and properties of these immense numbers.

Since the customary decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, a billion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (10) when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 (10) in exponential. Values that vary dramatically can be represented and compared graphically via logarithmic scale.

Billion is a word for a large number, and it has two distinct definitions:

• 1,000,000,000, i.e. one thousand million, or 10 (ten to the ninth power), as defined on the short scale. This is now the most common sense of the word in all varieties of English; it has long been established in American English and has since become common in Britain and other English-speaking countries as well.
• 1,000,000,000,000, i.e. one million million, or 10 (ten to the twelfth power), as defined on the long scale. This number is the historical sense of the word and remains the established sense of the word in other European languages. Though displaced by the short scale definition relatively early in US English, it remained the most common sense of the word in Britain until the 1950s and still remains in occasional use there.

American English adopted the short scale definition from the French (it enjoyed usage in France at the time, alongside the long-scale definition). The United Kingdom used the long scale billion until 1974, when the government officially switched to the short scale, but since the 1950s the short scale had already been increasingly used in technical writing and journalism. Moreover, in 1941, Winston Churchill remarked: "For all practical financial purposes a billion represents one thousand millions..."

Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.

Arithmetic systems can be distinguished based on the type of numbers they operate on. Integer arithmetic is about calculations with positive and negative integers. Rational number arithmetic involves operations on fractions of integers. Real number arithmetic is about calculations with real numbers, which include both rational and irrational numbers.

In mathematics, a basic algebraic operation is a mathematical operation similar to any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). The operations of elementary algebra may be performed on numbers, in which case they are often called arithmetic operations. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields.

An algebraic operation on a set ⁠${\displaystyle S}$⁠ may be defined more formally as a function that maps to ⁠${\displaystyle S}$⁠ the tuples of a given length of elements of ⁠${\displaystyle S}$⁠. The length of the tuples is called the arity of the operation, and each member of the tuple is called an operand. The most common case is the case of arity two, where the operation is called a binary operation and the operands form an ordered pair. A unary operation is an operation of arity one that has only one operand; for example, the square root. An example of a ternary operation (arity three) is the triple product.

Exponential growth occurs when a quantity grows as an exponential function of time. The quantity grows at a rate directly proportional to its present size. For example, when it is 3 times as big as it is now, it will be growing 3 times as fast as it is now.

In more technical language, its instantaneous rate of change (that is, the derivative) of a quantity with respect to an independent variable is proportional to the quantity itself. Often the independent variable is time. Described as a function, a quantity undergoing exponential growth is an exponential function of time, that is, the variable representing time is the exponent (in contrast to other types of growth, such as quadratic growth). Exponential growth is the inverse of logarithmic growth.