## Real numbers

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## Real Numbers

# Real Numbers

A real number is any number which can be represented by a point on the number line. The numbers 3.5, 0.003, 2/3, π, and are all real numbers.

The real numbers include the rational numbers, which are those which can be expressed as the ratio of two integers, and the irrational numbers, which cannot. (In the list above, all the numbers except π and are rational.)

It is thought that the first real number to be identified as irrational was discovered by the Pythagoreans in the sixth century BC. Prior to this discovery, people believed that every number could be expressed as the ratio of two natural numbers (negative numbers had not been discovered yet). The Pythagoreans were able to show, however, that the hypotenuse of an isosceles right triangle could not be measured exactly by any scale, no matter how fine, which would exactly measure the legs.

To see what this means, imagine a number line with an isosceles right triangle drawn upon it, as in Figure 1. Imagine that the legs are one unit long.

The Pythagoreans were able to show that no matter how finely each unit was subdivided (uniformly), point P would fall somewhere inside one of those subdivisions. Even if there were a million, a billion, a billion and one, or any other number of uniform subdivisions, point P would be missed by every one of them. It would fall inside a subdivision, not at an end. Point P represents a real number because it is a definite point on the number line, but it does not represent any rational numbera /b.

Point P is not the only irrational point. The square root of any prime number is irrational. So is the cube root, or any other root. In fact, by using infinite decimals to represent the real numbers, the mathematician Cantor was able to show that the number of real numbers is uncountable. An infinite set of numbers is “countable” if there is some way of listing them that allows one to reach any particular one of them by reading far enough down the list. The set of natural numbers is countable because the ordinary counting process will, if it is continued long enough, bring one to any particular number in the set. In the case of the irrational numbers, however, there are so many of them that every conceivable listing of them will leave at least one of them out.

The real numbers have many familiar subsets that are countable. These include the natural numbers, the integers, the rational numbers, and the algebraic numbers (algebraic numbers are those that can be roots of polynomial equations with integral coefficients). The real numbers also include numbers that are “none of the above.” These are the transcendental numbers, and they are uncountable. Pi is one.

Except for rare instances such as ÷ , computations can be done only with rational numbers. When one wants to use an irrational number such as π, , or e in a computation, one must replace it with a rational approximation such as 22/7, 1.73205, or 2.718. The result is never exact. However, one can always come as close to the exact real-number answer as one wishes. If the approximation 3.14 for π does not come close enough for the purpose, then 3.142, 3.1416, or 3.14159 can be used. Each gives a closer approximation.

## real numbers

**real numbers ( reals)** The numbers that allow a numerical quantity to be assigned to every point on an infinite line or continuum. Real numbers are thus used to measure and calculate

*exactly*the sizes of any continuous line segments or quantities. The development of a number system that meets these requirements has proved to be a long and complex process that reached a conclusion only in the 19th century. Establishing theoretical foundations for mathematical developments such as the calculus have involved sorting out subtle, conflicting, and inconsistent ideas about the reals (such as infinitesimals). The set of reals is infinite and not countable: there does not exist a method of making finite representations or codings of real numbers. Research on the foundations of the continuum continues – for instance on computation with the reals and on the uses of infinitesimals.

The real numbers, like the natural numbers, are one of the truly fundamental data types. Unlike the natural numbers, however, reals cannot be represented exactly in computations. They can be approximated to any degree of accuracy by rational numbers.

A real number can be defined in several ways, for example as the limit of a sequence of rational numbers. A real

*x*is represented by a sequence

*q*(0),

*q*(1),… of rational numbers that approximates

*x*in the sense that for any degree of accuracy ϵ there exists some natural number

*n*such that for all

*k*>

*n*, |

*q*(

*k*) –

*x*| < ϵA real number is a

*computable real number*if there is an algorithm that allows us to compute an approximation to the number to any given degree of accuracy. A real

*x*is computable if (a) there is an algorithm that lists a sequence

*q*(0),

*q*(1),… of rational numbers that converges to

*x*, and (b) there is an algorithm that to any natural number

*k*finds a natural number

*p*(

*k*) such thatfor all

*n*>

*p*(

*k*), |

*q*(

*n*) –

*x*| < 2

^{–}

*Most of the real numbers that we know and use come from solving equations (e.g. the algebraic numbers) and evaluating equationally defined sequences (e.g. e and π) and are computable. However, most real numbers are noncomputable.*

^{k}The approximations to real numbers used in computers must have finite representations or codings. In particular, there are gaps and separations between adjacent pairs of the real numbers that are represented (see model numbers). The separation may be the same between all numbers (fixed-point) or may vary and depend on the size of the adjacent values (floating-point). Some programming languages ignore this difference, describing floating-point numbers as “real”. Calculations with real numbers on a computer must take account of these approximations.

## Real Numbers

# Real numbers

A real number is any number which can be represented by a **point** on a number line. The numbers 3.5, −0.003, 2/3, π, and √2 are all real numbers.

The real numbers include the rational numbers, which are those which can be expressed as the **ratio** of two **integers** , and the irrational numbers, which cannot. (In the list above, all the numbers except **pi** and the **square root** of 2 are rational.)

It is thought that the first real number to be identified as irrational was discovered by the Pythagoreans in the sixth century b.c. Prior to this discovery, people believed that every number could be expressed as the ratio of two **natural numbers** (**negative** numbers had not been discovered yet). The Pythagoreans were able to show, however, that the hypotenuse of an isosceles right triangle could not be measured exactly by any scale, no matter how fine, which would exactly measure the legs.

To see what this meant, imagine a number line with an isosceles right triangle drawn upon it, as in Figure 1. Imagine that the legs are one unit long.

The Pythagoreans were able to show that no matter how finely each unit was subdivided (uniformly), point P would fall somewhere inside one of those subdivisions. Even if there were a million, a billion, a billion and one, or any other number of uniform subdivisions, point P would be missed by every one of them. It would fall inside a subdivision, not at an end. Point P represents a real number because it is a definite point on the number line, but it does not represent any **rational number** a/b.

Point P is not the only irrational point. The square root of any prime number is irrational. So is the cube root, or any other root. In fact, by using infinite decimals to represent the real numbers, the mathematician Cantor was able to show that the number of real numbers is uncountable. An infinite set of numbers is "countable" if there is some way of listing them that allows one to reach any particular one of them by reading far enough down the list. The set of natural numbers is **countable** because the ordinary counting process will, if it is continued long enough, bring one to any particular number in the set. In the case of the irrational numbers, however, there are so many of them that every conceivable listing of them will leave at least one of them out.

The real numbers have many familiar subsets which are countable. These include the natural numbers, the integers, the rational numbers, and the algebraic numbers (algebraic numbers are those which can be roots of polynomial equations with **integral** coefficients). The real numbers also include numbers which are "none of the above." These are the **transcendental numbers** , and they are uncountable. Pi is one.

Except for rare instances such as √2 ÷ √8 , computations can be done only with rational numbers. When one wants to use an **irrational number** such as π, √3 , or e in a computation, one must replace it with a rational **approximation** such as 22/7, 1.73205, or 2.718. The result is never exact. However, one can always come as close to the exact real-number answer as one wishes. If

the approximation 3.14 for π does not come close enough for the purpose, then 3.142, 3.1416, or 3.14159 can be used. Each gives a closer approximation.

## Numbers, Real

# Numbers, Real

A real number line is a familiar way to picture various sets of numbers. For example, the divisions marked on a number line show the integers, which are the counting numbers {1, 2, 3,…}, with their opposites {−1, −2, −3,…}, with the number 0, which divides the positive numbers on the line from the negative numbers.

But what other numbers are on a real number line? One could make marks for all the fractions, such as , and so forth, as well as marks for all the decimal fractions, such as 0.1, −0.01, 0.0000001, and so on. Any number that can be written as the ratio of two integers (such as , , and so on), where the divisor is not 0, is called a rational number, and all the rational numbers are on a real number line.

Are there any other kinds of numbers on a real number line in addition to the integers and rational numbers? What about , which is approximately 1.4142? Because the decimal equivalent for never ends and never repeats, it is known as an irrational number. The set of real numbers consists of the integers and the rational numbers as well as the irrational numbers. Every real number corresponds to exactly one point on a real number line, and every point on a number line corresponds to exactly one real number.

Are there any "unreal" numbers? That is, are there any numbers that are not on a real number line? The set of real numbers is infinitely large, and one might think that it contains all numbers, but that is not so. For example, the solution to the equation *x* ^{2} = −1 does not lie on a real number line. The solution to that equation lies on another number line called an imaginary number line, which is usually drawn at right angles to a real number line.

There are numbers, called complex numbers, that are the sum of a real number and an imaginary number and are not found on either a real or an imaginary number line. These complex numbers are found in an area called the complex plane. So both imaginary and complex numbers are "unreal," so to speak, because they do not lie on a real number line.

The set of real numbers has several interesting properties. For example, when any two real numbers are added, subtracted, multiplied, or divided (excluding division by zero), the result is always a real number. Therefore, the set of real numbers is called "closed" for these four operations.

Similarly, the real numbers have the commutative, associative, and distributive properties. The real numbers also have an identity element for addition (0) and for multiplication (1) and inverse elements for all four operations. These properties, taken all together, are called the field properties, and the real numbers thus make up a field, mathematically speaking.

see also Field Properties; Integers; Number Sets; Numbers, Complex; Numbers, Irrational; Numbers, Rational; Numbers, Whole; Number System, Real.

*Lucia McKay*

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