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Resources tagged with Rational and irrational numbers similar to Infinite Continued Fractions:

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Broad Topics > Numbers and the Number System > Rational and irrational numbers

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Approximations, Euclid's Algorithm & Continued Fractions

Stage: 5

This article sets some puzzles and describes how Euclid's algorithm and continued fractions are related.

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Good Approximations

Stage: 5 Challenge Level: Challenge Level:1

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

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Impossible Square?

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Can you make a square from these triangles?

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Continued Fractions II

Stage: 5

In this article we show that every whole number can be written as a continued fraction of the form k/(1+k/(1+k/...)).

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The Square Hole

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

If the yellow equilateral triangle is taken as the unit for area, what size is the hole ?

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Proof Sorter - the Square Root of 2 Is Irrational

Stage: 5 Challenge Level: Challenge Level:1

Try this interactivity to familiarise yourself with the proof that the square root of 2 is irrational. Sort the steps of the proof into the correct order.

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An Introduction to Proof by Contradiction

Stage: 4 and 5

An introduction to proof by contradiction, a powerful method of mathematical proof.

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An Introduction to Irrational Numbers

Stage: 4 and 5

Tim Rowland introduces irrational numbers

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Irrational Arithmagons

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Can you work out the irrational numbers that belong in the circles to make the multiplication arithmagon correct?

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The Clue Is in the Question

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

This problem is a sequence of linked mini-challenges leading up to the proof of a difficult final challenge, encouraging you to think mathematically. Starting with one of the mini-challenges, how. . . .

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What Are Numbers?

Stage: 2, 3, 4 and 5

Ranging from kindergarten mathematics to the fringe of research this informal article paints the big picture of number in a non technical way suitable for primary teachers and older students.

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Be Reasonable

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Prove that sqrt2, sqrt3 and sqrt5 cannot be terms of ANY arithmetic progression.

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Equal Equilateral Triangles

Stage: 4 Challenge Level: Challenge Level:1

Using the interactivity, can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

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Rational Roots

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Given that a, b and c are natural numbers show that if sqrt a+sqrt b is rational then it is a natural number. Extend this to 3 variables.

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Road Maker 2

Stage: 5 Short Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you work out where the blue-and-red brick roads end?

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Spirostars

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

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The Root Cause

Stage: 5 Challenge Level: Challenge Level:1

Prove that if a is a natural number and the square root of a is rational, then it is a square number (an integer n^2 for some integer n.)

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Rational Round

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

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Rationals Between

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What fractions can you find between the square roots of 56 and 58?

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Impossible Triangles?

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Which of these triangular jigsaws are impossible to finish?