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Extend the Irrationality Proof to Other Square Roots

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The World of Numbers · Irrational lengths and proofs · 4:00

You'll learn: Adapt the contradiction method and identify the divisibility step needed.

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Extend the Irrationality Proof to Other Square Roots

4:00

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Extend the Irrationality Proof to Other Square Roots

You'll learn: Adapt the contradiction method and identify the divisibility step needed.

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Same topic

Irrational lengths and proofs

6

A Unit Square Reveals a New Kind of Number
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A Unit Square Reveals a New Kind of Number
You'll learn: Use the diagonal of a unit square to motivate √2 and irrational numbers.
Why √2 Cannot Be a Fraction
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Why √2 Cannot Be a Fraction
You'll learn: Follow the contradiction proof from a lowest-terms fraction to both numerator and denominator being even.
Construct √2 and Transfer It to the Number Line
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Construct √2 and Transfer It to the Number Line
You'll learn: Build a right triangle and use a compass arc to locate the exact irrational length.
Build More Square Roots with a Square Root Spiral
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Build More Square Roots with a Square Root Spiral
You'll learn: Extend successive unit-perpendicular triangles and identify hypotenuse lengths.
Pi: An Exact Number with Fractional Approximations
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Pi: An Exact Number with Fractional Approximations
You'll learn: Define π as circumference divided by diameter and distinguish exact π from a rational approximation.
How Can Adding More Terms Approach Pi?
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How Can Adding More Terms Approach Pi?
You'll learn: Explore the chapter’s alternating infinite-series example through finite partial sums.

Next topic

Real numbers and decimal expansions

9

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
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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?
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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
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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.