In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by a boldface Q (or blackboard bold , Unicode ℚ); it was thus denoted in 1895 by Giuseppe Peano after quoziente, Italian for "quotient".
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).
What is an irrational number?
A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating.
Video: Recognizing Rational and Irrational Numbers
Identifying rational numbers
This video shows how to identify rational numbers.
Equality of rational numbers
Video: Comparing rational numbers
if and only if
If both fractions are in canonical form then
if and only if and
Ordering of rational numbers
If both denominators are positive, and, in particular, if both fractions are in canonical form,
if and only if
If either denominator is negative, each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator by changing the signs of both its numerator and denominator.
Video: Ordering rational numbers
Addition of rational numbers
Two fractions are added as follows:
If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.
Subtraction of rational numbers
If both fractions are in canonical form, the result is in canonical form if and only if b and d are coprime integers.
Video: Adding and subtracting rational expressions
Multiplication of rational numbers
The rule for multiplication is:
Even if both fractions are in canonical form, the result may be a reducible fraction.
Video: Multiplication of rational numbers
Inverse of rational numbers
Every rational number a/b has an additive inverse, often called its opposite,
If a/b is in canonical form, the same is true for its opposite.
A nonzero rational number a/b has a multiplicative inverse, also called its reciprocal,
If a/b is in canonical form, then the canonical form of its reciprocal is either or , depending on the sign of a.
Video: Multiplicative and additive inverse of rational numbers
Division of rational numbers
If both b and c are nonzero, the division rule is
Thus, dividing a/b by c/d is equivalent to multiplying a/b by the reciprocal of c/d:
Video: Multiplying and Dividing Rational Numbers
Exponentiation to integer power
If n is a non-negative integer, then
The result is in canonical form if the same is true for a/b. In particular,
If a ≠ 0, then
If a/b is in canonical form, the canonical form of the result is if either a > 0 or n is even. Otherwise, the canonical form of the result is
Video: Basic fractional exponents
Finding irrational numbers between two rational numbers
This video shows how to find irrational numbers between two rational numbers.
Proof that the sum of rational and irrational numbers is also irrational
This video shows proof that the sum of rational and irrational numbers is also irrational.
Let's Review
A ___number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a ___number.
A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without ___.
Answer
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number.
A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating.