Properties of Rational Numbers Explained With Examples

Rational numbers are numbers that can be written as a ratio or fraction, where the numerator and denominator are integers. They are comprised of all of the counting numbers (1, 2, 3…), integers (…-3, -2, -1, 0, 1, 2…), and fractions. Rational numbers have distinct properties that govern how they relate to each other and how they can be manipulated.

In this article, we will explore the key properties of rational numbers in depth, including closure, commutative, associative, identity, inverse, and distributive properties. Examples will be provided to illustrate each property. Understanding these properties is essential for working with and comparing rational numbers efficiently.

1. Closure Property

The closure property of rational numbers states that performing arithmetic operations on rational numbers will always result in another rational number. The four basic arithmetic operations are addition, subtraction, multiplication, and division. Let’s look at some examples:

Addition:

1/2 + 3/4 = 5/4

The sum of two rational numbers is another rational number.

Subtraction:

2/3 – 1/6 = 1/2

The difference between two rational numbers is another rational number.

Multiplication:

1/3 x 2/5 = 2/15

The product of two rational numbers is another rational number.

Division:

3/4 ÷ 1/2 = 3/2

The quotient of two rational numbers is another rational number.

As you can see, closure applies to all four arithmetic operations for rational numbers. This allows us to generate more rational numbers from existing rational numbers.

2. Commutative Property

The commutative property states that changing the order of numbers does not affect the result when you add or multiply rational numbers. Here are some examples:

Addition:

Changing the order of the addends does not affect the sum.

Multiplication:

Changing the order of the factors does not affect the product.

Therefore, addition and multiplication of rational numbers are commutative. However, subtraction and division are not commutative, for example:

Subtraction:

Changing the order changes the difference.

Division:

Changing the order changes the quotient.

3. Associative Property

The associative property allows us to regroup rational numbers in different ways when adding or multiplying without changing the result. See the examples below:

Addition:

Regrouping the addends does not affect the sum.

Multiplication:

Regrouping the factors does not affect the product.

However, subtraction and division are not associative. Regrouping rational numbers when subtracting or dividing can change the end result.

4. Identity Property

The additive and multiplicative identity properties state that there are rational numbers that do not change other rational numbers when used in arithmetic operations.

The additive identity is 0. When 0 is added to any rational number, the rational number remains unchanged.

Examples:

The multiplicative identity is 1. When 1 is multiplied by any rational number, the rational number remains unchanged.

Examples:

These identities allow operations on rational numbers to be simplified or evaluated easily.

5. Inverse Property

Every rational number has an additive inverse and a multiplicative inverse. The additive inverse, or opposite, of a rational number is the number that can be added to the original to get 0. The multiplicative inverse of a non-zero rational number is the number that can be multiplied with the original to get 1.

For any rational number a/b:

Knowing how to find inverses allows you to “undo” operations involving rational numbers.

6. Distributive Property

The distributive property is very useful for multiplying rational numbers efficiently. It states that multiplying a rational number by a sum or difference is the same as multiplying each part separately and then adding or subtracting the products.

For rational numbers a/b, c/d, and e/f:

a/b x (c/d ± e/f) = a/b x c/d ± a/b x e/f

Where ± means you can use either the addition or subtraction symbol.

Examples:

• 2/3 x (1/4 + 3/5) = 2/3 x 1/4 + 2/3 x 3/5
  = 2/12 + 6/15
  = 10/60 + 24/60
  = 34/60
  = 17/30
• 1/2 x (3/4 – 1/6) = 1/2 x 3/4 – 1/2 x 1/6
  = 3/8 – 1/12
  = 9/24 – 2/24
  = 7/24

Using the distributive property makes multiplying rational expressions more efficient by breaking them into simpler parts.

7. Trichotomy Property

The trichotomy property states that given any two distinct rational numbers, one and only one of the following must be true:

1. a/b > c/d
2. a/b = c/d
3. a/b < c/d

This property allows us to unambiguously compare and order any two rational numbers. For example:

• 2/3 > 1/2 (2/3 is greater than 1/2)
• -3/4 = -3/4 (-3/4 is equal to itself)
• 1/5 < 4/9 (1/5 is less than 4/9)

Because of the trichotomy property, rational numbers can be arranged in order on the number line.

8. Properties of Equality

There are several properties governing equality of rational numbers:

1. Reflexive property: a/b = a/b. Any rational number is equal to itself.
2. Symmetric property: If a/b = c/d, then c/d = a/b. Equality works both ways.
3. Transitive property: If a/b = c/d and c/d = e/f, then a/b = e/f. Equality can be transferred along a sequence of equal rational numbers.
4. Substitution property: If a/b = c/d, then a/b can be substituted for c/d in any expression.

For example:

• Reflexive: 3/5 = 3/5
• Symmetric: If 4/6 = 2/3, then 2/3 = 4/6
• Transitive: If 1/2 = 2/4 and 2/4 = 4/8, then 1/2 = 4/8
• Substitution: If 3/7 = 6/14, then 3/7 + 1/2 = 6/14 + 1/2

These properties allow us to make logical conclusions regarding rational number equality.

9. Order Properties

Order properties relate to the way rational numbers are arranged from least to greatest value:

1. If a/b < c/d and c/d < e/f, then a/b < e/f. (Transitive property of inequality)
2. If a/b < c/d, then -a/b > -c/d.
3. If a/b < c/d, then a/b + e/f < c/d + e/f.
4. If a/b < c/d and e/f > 0, then a/b x e/f < c/d x e/f.

Examples:

1. 1/8 < 1/4 and 1/4 < 3/8, so 1/8 < 3/8
2. -3/7 < -1/6, so 3/7 > 1/6
3. 1/3 < 3/5, so 1/3 + 1/2 < 3/5 + 1/2
4. 1/10 < 3/7 and 4/5 > 0, so 1/10 x 4/5 < 3/7 x 4/5

These properties allow us to compare and order rational numbers.

10. Applications of Properties

Understanding the properties of rational numbers has many useful applications:

• Simplifying expressions with multiple operations by using properties like the distributive, commutative, and associative rules. This makes problems easier to solve.
• Determining if two rational numbers are equal by using properties like reflexivity, symmetry, and transitivity.
• Comparing and ordering rational numbers using properties like trichotomy and order properties. This allows you to arrange numbers on a number line.
• Finding additive inverses and multiplicative inverses using the inverse property. This helps solve equations involving rational numbers.
• Knowing that arithmetic on rational numbers will always result in another rational number due to the closure property. This ensures operations on rationals stay within the system.
• Substituting equivalent rational number fractions into expressions using the substitution property. This enables simplification of complex fractions.

Mastering these properties provides the tools needed for success with rational number arithmetic, equations, inequalities, representations on the number line, and more complex math concepts built on rational numbers.

11. Conclusion

In summary, rational numbers have many important properties that govern arithmetic relationships and comparisons between them. The closure, commutative, associative, identity, inverse, distributive, trichotomy, equality, order, and other properties of rational numbers all play a role in working efficiently and accurately with fractions, decimals, and ratios. Understanding these properties helps develop intuition for how rational numbers behave in various situations. With a solid grasp of rational number properties, moving on to more advanced mathematics topics like algebra will be easier. The properties provide the rules of the game for working with these incredibly useful numbers.