The World of Numbers · Lesson 4 of 7
Filling the Spaces: Fractions and Rational Numbers
“Fractions prove that numbers do not have to be whole to be useful.”
• Understand fractions and negative fractions as parts of a whole. • Define rational numbers in the form p/q. • Understand equivalent rational numbers and lowest form. • Apply arithmetic laws for rational numbers. • Represent rational numbers on the number line and understand density.
Counting is useful when we are dealing with complete objects, such as 4 books, 7 chairs, or 12 students. But measurement is not always so neat. Length, time, mass, money, and many other quantities can fall between whole numbers, so integers alone are not enough.
Fractions help us describe these in-between values accurately. They can represent parts of a whole, such as one-half of a cup or three-fourths of a field, but they can also represent quantities greater than one, such as five-thirds of a metre, or quantities less than zero, such as a debt of three-fourths of a unit. In this way, fractions extend the number system and allow us to describe measurements and quantities much more precisely.
A fraction represents a part of a whole or a ratio between quantities, commonly written in the form p/q.
Just as positive integers have negative counterparts, positive fractions also have additive inverses. For example, the additive inverse of 3/4 is −3/4.
Rational Numbers
A number that can be expressed as p/q, where p and q are integers and q ≠ 0.
Integers are also rational because every integer can be written with denominator 1. For example, 5 = 5/1 and −10 = −10/1. Thus the rational numbers contain the natural numbers, whole numbers and integers.
Why q Cannot Be Zero
Division asks how many groups of the divisor fit into the dividend. Division by zero does not produce a meaningful finite number because zero groups cannot be used to build a non-zero quantity. Therefore q = 0 is excluded from the definition.
Equivalent Rational Numbers
A rational number has many equivalent fraction forms. Multiplying or dividing numerator and denominator by the same non-zero integer does not change the value.
Problem
Reduce 12/30 to lowest form.
- 1.The greatest common factor of 12 and 30 is 6.
- 2.Divide numerator and denominator by 6.
- 3.12/30 = 2/5.
- 4.Since 2 and 5 have no common factor other than 1, 2/5 is in lowest form.
Equality of Rational Numbers
Problem
Verify using cross multiplication.
- 1.For 2/3 and 4/6, calculate 2 × 6 = 12.
- 2.Calculate 3 × 4 = 12.
- 3.The cross products are equal.
- 4.Therefore 2/3 = 4/6.
Arithmetic of Rational Numbers
For addition and subtraction with different denominators, first rewrite the fractions using a common denominator. Multiplication is direct: multiply numerators together and denominators together. Division is performed by multiplying by the reciprocal of the divisor.
Closure, Commutativity and Distributivity
Rational numbers are closed under addition, subtraction and multiplication. They are also closed under division as long as the divisor is not zero. Addition and multiplication are commutative, and multiplication distributes over addition.
Rational Numbers on the Number Line
A rational number p/q can be located by dividing a unit interval into q equal parts and moving p such parts from zero. Positive values lie to the right and negative values lie to the left.
Problem
How is 3/4 located between 0 and 1?
- 1.Divide the interval from 0 to 1 into 4 equal parts.
- 2.Each part has length 1/4.
- 3.Starting at 0, move 3 such parts to the right.
- 4.The point reached represents 3/4.
Problem
Where does 9/4 lie?
- 1.9/4 = 2 1/4.
- 2.So the point lies between 2 and 3.
- 3.Divide the interval from 2 to 3 into 4 equal parts.
- 4.Move one part to the right of 2.
- 5.That point is 9/4.
Absolute Value and Distance
The absolute value |x| is the distance of x from 0 on the number line. It is always non-negative.
The Density of Rational Numbers
Rational numbers are dense: between any two distinct rational numbers there is always another rational number. One convenient method is to take their average.
Problem
Find a rational number strictly between 1 and 3/2.
- 1.Take the average: [1 + 3/2]/2.
- 2.1 = 2/2, so the numerator becomes 5/2.
- 3.(5/2) ÷ 2 = 5/4.
- 4.Thus 5/4 lies between 1 and 3/2.
The same averaging process can be repeated again and again, which shows that infinitely many rational numbers lie between any two rational numbers.
Practice Problems
- Verify that 5/4 and 10/8 are equal.
- Find 2/5 + 3/10.
- Find 11/8 − 3/4.
- Find (−4/7) × (5/14).
- Find (2/3) ÷ (3/10).
- Represent 2/3, −5/4 and 11/2 on one number line.
- Find three rational numbers strictly between −1/2 and 1/4.
- A tailor has 15 3/4 m of silk and each kurta uses 2 1/4 m. Find the maximum number of complete kurtas that can be made.
- Find three rational numbers between 3.1415 and 3.1416.
Key Takeaways
• Rational numbers have the form p/q with integer p, q and q ≠ 0. • Integers are contained within the rational numbers. • Equivalent fractions represent the same rational number. • Rational arithmetic follows consistent laws for addition, subtraction, multiplication and division. • Rational numbers can be placed between integers on the number line. • Between any two rational numbers there are infinitely many more rational numbers.
Next, we discover that rational numbers are still not enough: some lengths cannot be written as any fraction at all.