Introduction to Rational Numbers
Rational numbers are an important part of mathematics, as they form the basis for arithmetic, algebra, and also real applications. They are used in performing basic operations such as addition, subtraction, multiplication, and division logically and without inconsistency. The word 'rational' originated from the word ratio, and in mathematics, rational numbers are represented in the form of a ratio of integers. That is, they are expressed in the form of 'p/q' where p and q are integers and q is not equal to 0. On the other hand, the numbers that cannot be represented in this form are called irrational numbers.
Rational numbers consist of whole numbers, integers, proper and improper fractions, as well as terminating and recurring decimals. Let's learn in detail about the properties, representation and application of rational numbers along with some easy-to-solve sample problems.
A rational number is any number that can be written in the form of a fraction in which the numerator and denominator are both integers and the denominator is not equal to zero. Notably, rational numbers are positive, negative, or zero and are densely ordered on the number line such that there is always another rational number between any two. Some examples of rational numbers are 1/3 or 0.333, whereas √7 is an irrational number since it cannot be expressed as an exact fraction and does not repeat or end. In contrast, √4 is a rational number since it reduces to 2.
Standard Form:
A number is rational if it can be expressed as p/q, where:
'p' is an integer (positive, negative, or zero)
'q' is a non-zero integer (q ≠ 0)
3/4 → rational (both 3 and 4 are integers, 4 ≠ 0)
-5 → rational (can be expressed as -5/1)
0 → rational (0/1)
1.5 → rational (can be expressed as 3/2)
-0.25 → rational (can be expressed as -1/4)
Rational numbers possess the following features:
They are positive, negative, or zero.
They can be expressed as terminating or recurring decimals.
They can be placed on the number line.
The collection of rational numbers is infinite.
Between any two rational numbers, there is another rational number (density property).
Rational numbers can be expressed in various forms, depending on how they are written or used. The following are the key types:
Every integer is a rational number, since it can be expressed as a ratio with a denominator of 1.
Examples:
-3 = -3/1
0 = 0/1
7 = 7/1
A proper fraction has a numerator less than the denominator.
Examples:
2/5
-3/8
1/10
An improper fraction has a numerator that is greater than or equal to the denominator.
Examples:
5/3
-9/4
12/4
A mixed number consists of a whole number and a fraction.
Examples:
1 1/2 = 3/2
3 3/4 = 15/4
-2 2/3 = -8/3
These decimals terminate or end after a finite number of digits.
Examples:
0.5 = 1/2
2.75 = 11/4
4.125 = 33/8
These decimals repeat one or more digits infinitely.
Examples:
0.333. = 1/3
0.666. = 2/3
1.272727. = 127/99
To determine whether a number is rational, ask the following:
1. Is the number expressible as p/q with integers p and q ≠ 0?
2. Does the decimal form terminate or repeat?
Examples:
0.25 → Yes (25/100 = 1/4)
-7 → Yes (-7/1)
0.666. → Yes (2/3)
√2 → No (non-repeating and non-terminating → irrational)
π → No (infinite and non-repeating → irrational)
Rational numbers can be represented on a number line between integers.
You can show them as:
Example:
To mark 3/4 on the number line, divide the region between 0 and 1 into 4 equal segments and mark the third segment.
Rational numbers may be added, subtracted, multiplied, and divided according to arithmetic operations.
Equate denominators (common denominator), then add/subtract numerators.
Example:
1/3 + 1/6 = (2/6) + (1/6) = 3/6 = 1/2
Multiply the numerators and denominators directly.
Example:
2/5 × 3/4 = (2×3)/(5×4) = 6/20 = 3/10
Divide by multiplying by the reciprocal of the divisor.
Example:
(2/3) ÷ (4/5) = (2/3) × (5/4) = 10/12 = 5/6
Rational numbers come in handy every day in daily life:
Amounts such as 1/2 cup sugar or 3/4 teaspoon salt
Interest rates, taxes, discounts (e.g., 6.5% = 13/200)
Percentages are sensible (e.g., 25% = 25/100 = 1/4)
Scores, averages, win-loss ratios
Measurements in inches, meters (e.g., 5 1/2 feet)
Rational numbers are closed under addition, subtraction, multiplication, and division (except by 0).
(a + b) + c = a + (b + c)
Commutative Property:
a + b = b + a
a × b = b × a
a × (b + c) = a × b + a × c
0 is the identity for addition: a + 0 = a
1 is the identity for multiplication: a × 1 = a
Additive inverse: a + (-a) = 0
Multiplicative inverse (reciprocal): a × (1/a) = 1 (a ≠ 0)
Rational numbers form the basis of arithmetic and problem-solving in real life. They are applicable in performing basic operations like addition, subtraction, multiplication, and division (excluding division by zero). They are widely used everywhere in our lives, from kitchen measurements and monetary calculations to building work, sales, and sporting statistics. Being familiar with rational numbers is crucial not just to master math but also to understand and handle much of daily life with reason and precision. Learn how to identify the difference between rational and irrational numbers and confidently work with them by practicing with our fun exercises and examples.
Ans: A rational number is any number that can be expressed in the form p/q, where:
p and q are integers
q ≠ 0
This includes:
Whole numbers
Fractions
Terminating decimals
Repeating decimals
Examples:
3/4 (both 3 and 4 are integers)
-5 (can be written as -5/1)
0.75 (can be written as 3/4)
2 (can be written as 2/1)
-7/2
Ans: Yes, 0.333333333... (repeating) is a rational number.
It is a repeating decimal, and all repeating decimals are rational because they can be written as a fraction.
In this case:
0.333... = 1/3
Ans: No, √7 is not a rational number.
The square root of 7 is a non-terminating, non-repeating decimal, which makes it irrational.
Approximate value:
√7 ≈ 2.645751311... (goes on forever without repeating)
Ans: Yes, √4 is a rational number.
√4 = 2, and 2 is a whole number, which can also be written as 2/1 (a rational number).
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