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Lesson 6 of 7

The World of Numbers · Lesson 6 of 7

Real Numbers: Decimals and Cyclic Patterns

“Every real number gets a place on the line, even when its decimal refuses to behave.”

Learning Objectives

• Understand real numbers as the union of rational and irrational numbers. • Distinguish terminating, repeating and non-repeating decimals. • Predict when a rational decimal terminates from the denominator. • Convert terminating and repeating decimals into p/q form. • Explore cyclic numbers and non-unique decimal representations.

Rational numbers and irrational numbers together form the real numbers. Every point on the ordinary number line corresponds to a real number, and every real number has a decimal expansion. The behaviour of that decimal expansion tells us whether the number is rational or irrational.

Definition
Real Number

Any rational or irrational number is a real number. The set of real numbers is denoted by R.

Real numbersLaTeX
Nesting of number systems Natural numbers are contained within integers, integers are contained within rational numbers, and rational and irrational numbers are separate sets contained within the real numbers. Real Numbers ℝ Rational Numbers ℚ Numbers expressible as p/q, where q ≠ 0 Integers ℤ … −3, −2, −1, 0, 1, 2, 3 … Natural Numbers ℕ 1, 2, 3, 4, … Examples: 1/2, −3/4, 9/4 Irrational Numbers Cannot be written in the form p/q √2, π, √3 Non-terminating and non-repeating decimals Rational numbers + Irrational numbers together form the Real numbers
Number-system nesting

Rational Decimals: Two Possibilities

When a rational number p/q is converted to decimal form by division, the decimal either terminates or eventually repeats.

Definition
Terminating Decimal

A decimal expansion that stops after finitely many digits.

Definition
Repeating Decimal

A non-terminating decimal in which a digit or block of digits repeats forever.

Rational numberDecimalType
3/80.375Terminating
5/110.454545...Repeating
10/33.333...Repeating
11/120.91666...Eventually repeating

Why Repetition Must Occur

Consider long division by 7. At any stage, the remainder can only be 0, 1, 2, 3, 4, 5 or 6. If remainder 0 appears, the decimal terminates. If it never appears, there are only finitely many possible non-zero remainders, so eventually one remainder must occur again. Once the same remainder returns, the later division steps repeat in the same cycle, producing a repeating decimal.

Predicting Terminating Decimals

For a rational number p/q in lowest terms, the decimal terminates exactly when the denominator q has no prime factors other than 2 and 5. The reason is that powers of 10 have only the prime factors 2 and 5.

Terminating conditionLaTeX
Worked Example: 3/20

Problem
Predict whether 3/20 terminates and find the decimal.

  1. 1.20 = 2² × 5.
  2. 2.Its prime factors are only 2 and 5.
  3. 3.Therefore the decimal must terminate.
  4. 4.Multiply numerator and denominator by 5: 3/20 = 15/100 = 0.15.
Worked Example: 4/15

Problem
Predict whether 4/15 terminates.

  1. 1.15 = 3 × 5.
  2. 2.The denominator contains prime factor 3.
  3. 3.Therefore the decimal cannot terminate.
  4. 4.It must be non-terminating and repeating.

Converting Terminating Decimals into p/q

Worked Example: 0.35

Problem
Convert 0.35 into p/q form.

  1. 1.0.35 = 35/100.
  2. 2.Divide numerator and denominator by 5: 35/100 = 7/20.
  3. 3.Therefore 0.35 = 7/20.

Pure Repeating Decimals

A pure repeating decimal begins repeating immediately after the decimal point. The algebraic method works by shifting one full repeating block to the left of the decimal and subtracting.

Worked Example: 0.666...

Problem
Convert 0.666... into p/q form.

  1. 1.Let x = 0.666...
  2. 2.Multiply by 10: 10x = 6.666...
  3. 3.Subtract the first equation: 10x − x = 6.
  4. 4.So 9x = 6.
  5. 5.x = 6/9 = 2/3.
Worked Example: 0.454545...

Problem
Convert 0.454545... into p/q form.

  1. 1.Let x = 0.454545...
  2. 2.Two digits repeat, so multiply by 100: 100x = 45.454545...
  3. 3.Subtract x: 99x = 45.
  4. 4.x = 45/99 = 5/11.

General Repeating Decimals

A general repeating decimal has one or more non-repeating digits first, followed by a repeating block. First shift the non-repeating digits before the decimal point. Then shift one entire repeating cycle and subtract.

Worked Example: 0.1666...

Problem
Convert 0.1666... into p/q form.

  1. 1.Let x = 0.1666...
  2. 2.One digit is non-repeating, so multiply by 10: 10x = 1.666...
  3. 3.Now shift one repeating digit by multiplying by 10 again: 100x = 16.666...
  4. 4.Subtract: 100x − 10x = 15.
  5. 5.90x = 15.
  6. 6.x = 1/6.
Worked Example: 2.357777...

Problem
Convert 2.357777... into p/q form.

  1. 1.Let x = 2.357777...
  2. 2.Two digits, 35, are non-repeating. Multiply by 100: 100x = 235.7777...
  3. 3.One digit repeats. Multiply by 10 again: 1000x = 2357.7777...
  4. 4.Subtract: 900x = 2122.
  5. 5.x = 2122/900 = 1061/450.

The Magic of Cyclic Numbers

The repeating block of 1/7 is 142857. Multiplying 142857 by 1, 2, 3, 4, 5 and 6 produces the same six digits in different cyclic orders.

MultiplierProduct
1142857
2285714
3428571
4571428
5714285
6857142

Irrational Decimals

Irrational numbers have decimal expansions that never terminate and never repeat a fixed block forever. For example, √2 = 1.4142135623... and π = 3.1415926535....

The Surprise 0.999... = 1

Worked Example: Why 0.999... Equals 1

Problem
Use algebra to prove 0.999... = 1.

  1. 1.Let x = 0.999...
  2. 2.Multiply by 10: 10x = 9.999...
  3. 3.Subtract the first equation from the second: 9x = 9.
  4. 4.Therefore x = 1.
  5. 5.Since x was 0.999..., we have 0.999... = 1.

This shows that decimal representation is not always unique. A terminating decimal can also be written using repeating 9s: 2.47000... = 2.46999....

A Glimpse Beyond the Real Numbers

The chapter ends by asking for √−1. No real number squares to −1, so √−1 does not lie on the real number line. This leads to imaginary numbers, represented using i, which belong to a later stage of mathematics.

Practice Problems

Practice Problems
  1. Without long division, decide whether 7/20, 4/15 and 13/250 terminate or repeat.
  2. Perform long division for 1/13 and identify its repeating block. Compare with 2/13 and 3/13.
  3. Classify as rational or irrational: √81, √12, 0.333..., 0.1234512345..., 1.01001000100001....
  4. Prove algebraically that 0.999... = 1.
  5. Convert 0.35, 0.666..., 0.454545... and 0.1666... into p/q form.
  6. A rational number in lowest terms has denominator 2³ × 5. Determine whether its decimal terminates and explain why.

Key Takeaways

Key Takeaways

• Real numbers are the union of rational and irrational numbers. • Rational decimals terminate or repeat. • A reduced rational denominator containing only factors 2 and 5 gives a terminating decimal. • Repeating decimals can be converted to fractions by algebraic shifting and subtraction. • Irrational decimals never terminate and never repeat a fixed block. • Cyclic numbers reveal hidden structure inside repeating rational decimals. • Some numbers have two decimal representations, such as 1 = 0.999....

Coming Next

Next, we bring the whole chapter together and practise moving confidently among natural, integer, rational, irrational and real numbers.

Previous · Lesson 5

Irrational Numbers

Next · Lesson 7

Chapter Summary and Practice