### There are 20 results

Broad Topics >

Numbers and the Number System > Rational and irrational numbers

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

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

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

Can you make a square from these triangles?

##### Age 16 to 18

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

##### Age 14 to 18

Tim Rowland introduces irrational numbers

##### Age 14 to 18

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

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

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

##### 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.

##### 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/...)).

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

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

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

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

##### Age 16 to 18 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. . . .

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

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

##### 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.

##### 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 ?

##### 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.)

##### 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.

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

Which of these triangular jigsaws are impossible to finish?

##### 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.

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

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

##### 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?