A rational number is a specific type of real number. It can be expressed in the form of a fraction p/q, where p and q are integers, and q is not zero. Essentially, any fraction where both the numerator and the denominator are integers qualifies as a rational number, provided that the denominator is non-zero. When you break down a rational number into a decimal, it can either be terminating or repeating.
Are All Rational Numbers Integers?
No, every rational number is not an integer. A rational number is defined as any number that can be expressed in the form p/q, where p and q are integers and q is not zero. While all integers are rational numbers (because any integer n can be written as n/1), not all rational numbers are integers.
For example, 3/4 and -7/2 are rational numbers but not integers. They fit the definition of rational numbers because they can be expressed as a fraction of two integers, but they are not whole numbers themselves.
In summary, integers are a subset of rational numbers, but rational numbers include many values that are not integers.
Key Properties of Rational Numbers
- Closed Operations: Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero). This means that performing these operations on rational numbers will always yield another rational number.
- Consistency in Operations: If you multiply or divide both the numerator and denominator of a rational number by the same non-zero integer, the value of the rational number remains unchanged.
- Multiplication with Zero: Multiplying a rational number by zero results in zero.
- Equality and Comparisons: Rational numbers can be compared, and their relationships can be determined based on their numerators and denominators.
Examples of Rational Numbers
Here are some examples of rational numbers: 9/11, 12/7, 3/5, -1/5, and -2/17.
Finding Rational Numbers Between Given Values
Example 1: Find a Rational Number Between 5/11 and 8/11
Solution: Since the denominators are the same, we compare the numerators directly. Between the numerators 5 and 8, integers 6 and 7 fall. Thus, rational numbers between 5/11 and 8/11 include 6/11 and 7/11.
Example 2: Determine the Rational Numbers Between 1/2 and 2/3
Solution: To compare 1/2 and 2/3, first find the least common multiple (LCM) of the denominators 2 and 3, which is 6. Convert the fractions to have this common denominator:
1/2 × 3/3 = 3/6
2/3 × 2/2 = 4/6
The numerators are now 3 and 4. Rational numbers between 3/6 and 4/6 include those with denominators of 60 when multiplied by 10:
3/6 × 10/10 = 30/60
4/6 × 10/10 = 40/60
Thus, rational numbers between 1/2 and 2/3 include: 31/60, 32/60, 33/60, 34/60, 35/60, 36/60, 37/60, 38/60, and 39/60.