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

##### Age 11 to 14 Challenge Level:

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator? ### An Introduction to Irrational Numbers

##### Age 14 to 18

Tim Rowland introduces irrational numbers ### Irrational Arithmagons

##### Age 16 to 18 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

##### Age 16 to 18 Challenge Level:

Starting with one of the mini-challenges, how many of the other mini-challenges will you invent for yourself? ##### Age 16 to 18 Short Challenge Level:

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

##### Age 16 to 18 Challenge Level:

Which of these triangular jigsaws are impossible to finish? ### The Square Hole

##### Age 14 to 16 Challenge Level:

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

##### Age 14 to 16 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 ? ### Impossible Square?

##### Age 16 to 18 Challenge Level:

Can you make a square from these triangles? ### What Are Numbers?

##### Age 7 to 18

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

##### Age 16 to 18 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? ### An Introduction to Proof by Contradiction

##### Age 14 to 18

An introduction to proof by contradiction, a powerful method of mathematical proof. ### The Dangerous Ratio

##### Age 11 to 14

This article for pupils and teachers looks at a number that even the great mathematician, Pythagoras, found terrifying. ### All Is Number

##### Age 7 to 14 ### Rational Round

##### Age 16 to 18 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. ### Proof Sorter - the Square Root of 2 Is Irrational

##### Age 16 to 18 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. ### Approximations, Euclid's Algorithm & Continued Fractions

##### Age 16 to 18

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

##### Age 16 to 18

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

##### Age 11 to 14 Challenge Level:

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square? ### Rationals Between...

##### Age 14 to 16 Challenge Level:

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

##### Age 16 to 18 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

##### Age 16 to 18 Challenge Level:

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers. ### The Root Cause

##### Age 16 to 18 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.) ### Be Reasonable

##### Age 16 to 18 Challenge Level:

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