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Why the Digits 142857 Keep Rotating

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The World of Numbers · Real numbers and decimal expansions · 3:34

You'll learn: Explore a cyclic number arising from the repeating block of 1/7.

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Why the Digits 142857 Keep Rotating

3:34

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Why the Digits 142857 Keep Rotating

You'll learn: Explore a cyclic number arising from the repeating block of 1/7.

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Real numbers and decimal expansions

12

Rational and Irrational Numbers Share One Real Number Line
3:44
Rational and Irrational Numbers Share One Real Number Line
You'll learn: Connect natural numbers, integers, rationals, irrationals and reals in a nested classification.
Why Rational Decimals Either Stop or Repeat
3:30
Why Rational Decimals Either Stop or Repeat
You'll learn: Distinguish terminating decimals from repeating ones and use long division.
Repeated Remainders Make Repeated Digits
3:39
Repeated Remainders Make Repeated Digits
You'll learn: Explain why a long-division process eventually terminates or cycles.
Predict a Terminating Decimal Without Dividing
3:41
Predict a Terminating Decimal Without Dividing
You'll learn: Reduce a fraction first, then inspect prime factors of its denominator.
How Many Decimal Places Will a Fraction Need?
3:39
How Many Decimal Places Will a Fraction Need?
You'll learn: Balance powers of 2 and 5 to turn a reduced denominator into a power of ten.
Turn a Terminating Decimal into a Fraction
3:42
Turn a Terminating Decimal into a Fraction
You'll learn: Use place value to write a power-of-ten denominator and reduce.
Turn a Pure Repeating Decimal into a Fraction
3:32
Turn a Pure Repeating Decimal into a Fraction
You'll learn: Multiply by a power of ten matching the cycle length, subtract and solve.
A Non-Repeating Start Followed by One Repeating Digit
3:46
A Non-Repeating Start Followed by One Repeating Digit
You'll learn: Align repeating tails after moving the non-repeating prefix.
Convert a Mixed Decimal with a Repeating Block
3:50
Convert a Mixed Decimal with a Repeating Block
You'll learn: Extend tail alignment to a multi-digit repeating cycle and select the correct powers of ten.
Investigate the Repeating Blocks of Thirteenths
3:37
Investigate the Repeating Blocks of Thirteenths
You'll learn: Compute and compare repeating blocks for successive multiples of 1/13.
A Decimal Can Have a Pattern Without Repeating
3:55
A Decimal Can Have a Pattern Without Repeating
You'll learn: Classify rational and irrational numbers using exact form and decimal behaviour.
Why 0.999… Is Exactly 1
3:36
Why 0.999… Is Exactly 1
You'll learn: Use algebraic cancellation to prove equality and explain two decimal representations of terminating values.

Next topic

Closing perspective

1

Why √−1 Is Not on the Real Number Line
2:59
Why √−1 Is Not on the Real Number Line
You'll learn: Explain why squaring any real number cannot produce −1 and introduce the chapter’s final question.

Next topic

Chapter challenge problems

2

Find More Reciprocals with Cyclic Repeating Blocks
3:46
Find More Reciprocals with Cyclic Repeating Blocks
You'll learn: Test reciprocal decimals and compare rotations of their repeating blocks.
What Does the Fourth Decimal Place Tell Us?
3:38
What Does the Fourth Decimal Place Tell Us?
You'll learn: Relate a last non-zero fourth decimal digit to a numerator over 10⁴ and examine reduction.