A rational number is a real number that can be expressed exactly as the ratio of two integers.^{[1]}
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 |
p 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 |
Any finite
decimal
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 a_{1}, a_{2}, b_{1} and b_{2} are integers, a_{1}b_{2} and a_{2}b_{1} 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.
Property | Description |
---|---|
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 ℵ_{0}. |
Table 1: Properties of the rational numbers. |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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