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

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

Approximations, Euclid's Algorithm & Continued Fractions

Stage: 5

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

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/...)).

Rationals Between...

Stage: 4 Challenge Level:

What fractions can you find between the square roots of 65 and 67?

Rational Roots

Stage: 5 Challenge Level:

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.

Good Approximations

Stage: 5 Challenge Level:

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

An Introduction to Irrational Numbers

Stage: 4 and 5

Tim Rowland introduces irrational numbers

Irrational Arithmagons

Stage: 5 Challenge Level:

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

The Clue Is in the Question

Stage: 5 Challenge Level:

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. . . .

Impossible Triangles?

Stage: 5 Challenge Level:

Which of these triangular jigsaws are impossible to finish?

An Introduction to Proof by Contradiction

Stage: 4 and 5

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

Be Reasonable

Stage: 5 Challenge Level:

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

The Root Cause

Stage: 5 Challenge Level:

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.)

Stage: 5 Short Challenge Level:

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

Impossible Square?

Stage: 5 Challenge Level:

Can you make a square from these triangles?

Equal Equilateral Triangles

Stage: 4 Challenge Level:

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

The Square Hole

Stage: 4 Challenge Level:

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

Proof Sorter - the Square Root of 2 Is Irrational

Stage: 5 Challenge Level:

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.

Rational Round

Stage: 5 Challenge Level:

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

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.

Spirostars

Stage: 5 Challenge Level:

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?