An integer is always a rational number. This is because integers can be expressed as a ratio of themselves and 1. For example, the number 5 can be written as 5 / 1.
In math, numbers can be represented in ways that mask their true identity. For example, the square root of 4 can be reduced to 2, and so is a rational number, even if it is represented by using a square root.
|5.2||All finite decimals are rational numbers. Why?|
|All fractions with a rational numerator and denominator are rational numbers. Why?|
|3||All integers are rational numbers. Why?|
|Any representation of a number that can be simplified to a rational number is also a rational number.|
|3.420742074207…||Any repeating decimal can be represented as a fraction with integer numerator and denominator. So any repeating decimal is a rational number.|
|Table 1: Representations of rational numbers|
|π||π has been proven to be irrational.|
|Any square root that can not be simplified to a rational number is irrational.|
|Table 2: Representations of irrational numbers|
can be represented by a fraction of integers. Using the definition of a decimal number,
the number 5.2 can be represented as
Since all rational numbers can be represented as the ratio of two integers, the fraction can be written as where and . Using the properties of multiplication, Since a1, a2, b1 and b2 are integers, a1b2 and a2b1 are also integers, is a rational number.
Start with the fact that anything divided by one remains unchanged. So Since both 3 and 1 are integers, is a rational number, so 3 must also be a rational number.
The definition of a rational number is a number that can be represented as the ratio of two integers. If a square root can be simplified to a rational number, then that square root represents a rational number. Since , represents a rational number.
The repeating decimal 3.420742074207… can be written as . Since a repeating decimal can be written as the ratio of two integers, all repeating decimals are rational number.
|Associativity||The set of rational numbers is associative with respect to addition, subtraction, multiplication and division. Example: a + (b + c) = (a + b) + c.|
|Commutativity||The set of rational numbers is commutative with respect to addition and multiplication. The set of rational numbers is not commutative with respect to subtraction or division. Example: a + b = b + a.|
|Additive identity||The additive identity for rational numbers is 0. Example: a + 0 = 0 + a = a.|
|Multiplicative identity||The multiplicative identity for rational numbers is 1. Example: a · 1 = 1 · a = a.|
|Closure||The set of rational numbers is closed with respect to addition, subtraction, multiplication, and division. Example: if a and b are rational numbers then a + b is also a rational number.|
|Discrete||The set of rational numbers is a discrete (not continuous) set.|
|Cardinality||The cardinality of the set of rational numbers is ℵ0.|
|Table 1: Properties of the rational numbers.|
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