Rational Number

Rational Number

What is a rational number?

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

  1. 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.

  2. 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. 

  3. 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

  1. 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.

  2. 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. 

  3. 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.

 

Test Your Knowledge

 

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