The real numbers are all the numbers on a number line extending from negative infinity through zero to positive infinity. This construction of the set of real numbers is not arbitrary but rather the result of an evolution from the natural numbers used for counting. The system of natural numbers has several inconsistencies, and as calculations became more complex, the number system expanded to address its limitations. With real numbers, calculations give consistent results, and there are few exceptions or limitations such as were present with the more primitive versions of the number system.
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The set of real numbers consists of all the numbers on a number line. This includes natural numbers, whole numbers, integers, rational numbers and irrational numbers. It does not include imaginary numbers or complex numbers.
Natural Numbers and Closure
Closure is the property of a set of numbers that means if allowed calculations are performed on numbers that are members of the set, the answers will also be numbers that are members of the set. The set is said to be closed.
Natural numbers are the counting numbers, 1, 2, 3..., and the set of natural numbers is not closed. As natural numbers were used in commerce, two problems immediately arose. While the natural numbers counted real objects, for example cows, if a farmer had five cows and sold five cows, there was no natural number for the result. Early number systems very quickly developed a term for zero to address this problem. The result was the system of whole numbers, which is the natural numbers plus zero.
The second problem was also associated with subtraction. As long as numbers counted real objects such as cows, the farmer could not sell more cows than he had. But when numbers became abstract, subtracting larger numbers from smaller ones gave answers outside the system of whole numbers. As a result, integers, which are the whole numbers plus negative natural numbers were introduced. The number system now included a complete number line but only with integers.
Calculations in a closed number system should give answers from within the number system for operations such as addition and multiplication but also for their inverse operations, subtraction and division. The system of integers is closed for addition, subtraction and multiplication but not for division. If an integer is divided by another integer, the result is not always an integer.
Dividing a small integer by a larger one gives a fraction. Such fractions were added to the number system as rational numbers. Rational numbers are defined as any number that can be expressed as a ratio of two integers. Any arbitrary decimal number can be expressed as a rational number. For example 2.864 is 2864/1000 and 0.89632 is 89632/100,000. The number line now seemed to be complete.
There are numbers on the number line that cannot be expressed as a fraction of integers. One is the ratio of the sides of a right-angled triangle to the hypotenuse. If two of the sides of a right-angled triangle are 1 and 1, the hypotenuse is the square root of 2. The square root of two is an infinite decimal that does not repeat. Such numbers are called irrational, and they include all real numbers that are not rational. With this definition, the number line of all real numbers is complete because any other real number that is not rational is included in the definition of irrational.
Although the real number line is said to extend from negative to positive infinity, infinity itself is not a real number but rather a concept of the number system that defines it as being a quantity larger than any number. Mathematically infinity is the answer to 1/x as x reaches zero, but division by zero is not defined. If infinity were a number, it would lead to contradictions because infinity does not follow the laws of arithmetic. For example, infinity plus 1 is still infinity.
The set of real numbers is closed for addition, subtraction, multiplication and division except for division by zero, which is not defined. The set is not closed for at least one other operation.
The rules of multiplication in the set of real numbers specify that the multiplication of a negative and a positive number gives a negative number while the multiplication of positive or negative numbers gives positive answers. This means that the special case of multiplying a number by itself yields a positive number for both positive and negative numbers. The inverse of this special case is the square root of a positive number, giving both a positive and a negative answer. For the square root of a negative number, there is no answer in the set of real numbers.
The concept of the set of imaginary numbers addresses the issue of negative square roots in the real numbers. The square root of minus 1 is defined as i and all imaginary numbers are multiples of i. To complete number theory, the set of complex numbers is defined as including all real and all imaginary numbers. Real numbers can continue to be visualized on a horizontal number line while imaginary numbers are a vertical number line, with the two intersecting at zero. Complex numbers are points in the plane of the two number lines, each with a real and an imaginary component.