User:Phlsph7/Arithmetic - Numbers

Numbers
Numbers are mathematical objects used to count quantities and measure magnitudes. They are fundamental elements in arithmetic since all arithmetic operations are performed on numbers. There are different types of numbers and different numeral systems utilized to represent them.

Types


The main types of numbers employed in arithmetic are natural numbers, whole numbers, integers, rational numbers, and real numbers. The natural numbers are whole numbers that start from 1 and go to infinity. They exclude 0 and negative numbers. They are also known as counting numbers and can be expressed as {1, 2, 3, 4, ...}. The symbol of the natural numbers is $$\N$$. The whole numbers are identical to the natural numbers with the only difference being that they include 0. They can be represented as {0, 1, 2, 3, 4, ...} and have the symbol $$\N_0$$. Some mathematicians do not draw the distinction between the natural and the whole numbers by including 0 in the set of natural numbers. The set of integers encompasses both positive and negative whole numbers. It has the symbol $$\Z$$ and can be expressed as {..., -2, -1, 0, 1, 2, ...}.

A number is rational if it can be represented as the ratio of two integers. For example, the rational number $$\tfrac{1}{2}$$ is formed by dividing the integer 1, called the numerator, by the integer 2, called the denominator. Other examples are $$\tfrac{3}{4}$$ and $$\tfrac{281}{3}$$. The set of rational numbers includes all integers, which are fractions with a denominator of 1. The symbol of the rational numbers is $$\Q$$. Decimal fractions like 0.3 and 25.12 are a special type of rational numbers since their denominator is a power of 10. For example, 0.3 is equal to $$\tfrac{3}{10}$$, and 25.12 is equal to $$\tfrac{2512}{100}$$. Every rational number corresponds to a finite or a repeating decimal.

Irrational numbers are numbers that cannot be expressed through the ratio of two integers. Examples are many square roots, such as $$\sqrt 2$$, and numbers like $\pi$ and e (Euler's number). The decimal representation of an irrational number is infinite without repeating decimals. The set of rational numbers together with the set of irrational numbers makes up the set of real numbers. The symbol of the real numbers is $$\R$$ Even wider classes of numbers include complex numbers and quaternions.

Based on how numbers are used, they can be distinguished into cardinal and ordinal numbers. Cardinal numbers, like one, two, and three, are numbers that express the quantity of objects. They answer the question "how many?". Ordinal numbers, such as first, second, and third, indicate order or placement in a series. They answer the question "what position?".

Numeral systems
A numeral is a symbol to represent a number and numeral systems are representational frameworks. They usually have a limited amount of basic numerals, which directly refer to certain numbers. The system governs how these basic numerals may be combined to express any number. Numeral systems are either positional or non-positional. All early numeral systems were non-positional. For non-positional numeral systems, the value of a digit does not depend on its position in the combined numeral.

The simplest non-positional system is the unary numeral system. It relies on one symbol for the number 1. All higher numbers are written by repeating this symbol. For example, the number 7 can be represented by repeating the symbol for 1 seven times. This system makes it cumbersome to write large numbers, which is why many non-positional systems include additional symbols to directly represent larger numbers. Variations of the unary numeral systems are employed in tally sticks using dents and in tally marks.



Egyptian hieroglyphics had a more complex non-positional numeral system. They have additional symbols for numbers like 10, 100, 1000, and 10,000. These symbols can be combined into a sum to more conveniently express larger numbers. For example, the numeral for 10,405 uses one time the symbol for 10,000, four times the symbol for 100, and five times the symbol for 1. A similar well-known framework is the Roman numeral system. It has the symbols I, V, X, L, C, D, M as its basic numerals to represent the numbers 1, 5, 10, 50, 100, 500, and 1000.

A numeral system is positional if the position of a basic numeral in a compound expression determines its value. Positional numeral systems have a radix that acts as a multiplier of the different positions. For each subsequent position, the radix is raised to a higher power. In the common decimal system, also called the Hindu–Arabic numeral system, the radix is 10. This means that the first digit is multiplied by $$10^0$$, the next digit is multiplied by $$10^1$$, and so on. For example, the decimal numeral 532 stands for $$5 \cdot 10^2 + 3 \cdot 10^1 + 2 \cdot 10^0$$. Because of the effect of the digits' positions, the numeral 532 differs from the numerals 325 and 253 even though they have the same digits.

Another positional numeral system used extensively in computer arithmetic is the binary system, which has a radix of 2. This means that the first digit is multiplied by $$2^0$$, the next digit by $$2^1$$, and so on. For example, the number 13 is written as 1101 in the binary notation, which stands for $$1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0$$. In computing, each digit in the binary notation is corresponds to one bit. The earliest positional system was developed by ancient Sumerians and had a radix of 60.