Other tags that relate to A Brief Introduction to Complex Numbers
Limits of Sequences. Algorithms. Mathematical reasoning & proof. Arithmetic sequences. Proof by contradiction. Rational and irrational numbers. Video. Summation of series. Angular Motion. Practical Activity.There are 158 results
Broad Topics > Thinking Mathematically > Mathematical reasoning & proof
Seven Squares - Group-worthy Task
Age 11 to 14 Challenge Level:
Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?
Cyclic Quad Jigsaw
Age 14 to 16 Challenge Level:
A picture is made by joining five small quadrilaterals together to make a large quadrilateral. Is it possible to draw a similar picture if all the small quadrilaterals are cyclic?
More Marbles
Age 11 to 14 Challenge Level:
I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?
Air Nets
Age 7 to 18 Challenge Level:
Can you visualise whether these nets fold up into 3D shapes? Watch the videos each time to see if you were correct.
Tis Unique
Age 11 to 14 Challenge Level:
This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.
Picture Story
Age 14 to 16 Challenge Level:
Can you see how this picture illustrates the formula for the sum of the first six cube numbers?
Natural Sum
Age 14 to 16 Challenge Level:
The picture illustrates the sum 1 + 2 + 3 + 4 = (4 x 5)/2. Prove the general formula for the sum of the first n natural numbers and the formula for the sum of the cubes of the first n natural. . . .
Triangle Incircle Iteration
Age 14 to 16 Challenge Level:
Keep constructing triangles in the incircle of the previous triangle. What happens?
Archimedes and Numerical Roots
Age 14 to 16 Challenge Level:
The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?
Symmetric Tangles
Age 14 to 16
The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!
Marbles
Age 11 to 14 Challenge Level:
I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?
What Numbers Can We Make?
Age 11 to 14 Challenge Level:
Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?
Proximity
Age 14 to 16 Challenge Level:
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
More Number Pyramids
Age 11 to 14 Challenge Level:
When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
Take Three from Five
Age 14 to 16 Challenge Level:
Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?
Tower of Hanoi
Age 11 to 14 Challenge Level:
The Tower of Hanoi is an ancient mathematical challenge. Working on the building blocks may help you to explain the patterns you notice.
Rhombus in Rectangle
Age 14 to 16 Challenge Level:
Take any rectangle ABCD such that AB > BC. The point P is on AB and Q is on CD. Show that there is exactly one position of P and Q such that APCQ is a rhombus.
Paradoxes
Age 7 to 14
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Unit Fractions
Age 11 to 14 Challenge Level:
Consider the equation 1/a + 1/b + 1/c = 1 where a, b and c are natural numbers and 0 < a < b < c. Prove that there is only one set of values which satisfy this equation.
Pent
Age 14 to 18 Challenge Level:
The diagram shows a regular pentagon with sides of unit length. Find all the angles in the diagram. Prove that the quadrilateral shown in red is a rhombus.
Matter of Scale
Age 14 to 16 Challenge Level:
Prove Pythagoras' Theorem using enlargements and scale factors.
Children at Large
Age 11 to 14 Challenge Level:
There are four children in a family, two girls, Kate and Sally, and two boys, Tom and Ben. How old are the children?
Coins on a Plate
Age 11 to 14 Challenge Level:
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
Composite Notions
Age 14 to 16 Challenge Level:
A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.
Königsberg
Age 11 to 14 Challenge Level:
Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps?
Pattern of Islands
Age 11 to 14 Challenge Level:
In how many distinct ways can six islands be joined by bridges so that each island can be reached from every other island...
Similarly So
Age 14 to 16 Challenge Level:
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
Round and Round
Age 14 to 16 Challenge Level:
Prove that the shaded area of the semicircle is equal to the area of the inner circle.
Top-heavy Pyramids
Age 11 to 14 Challenge Level:
Use the numbers in the box below to make the base of a top-heavy pyramid whose top number is 200.
Number Rules - OK
Age 14 to 16 Challenge Level:
Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number...
Picturing Pythagorean Triples
Age 14 to 18
This article discusses how every Pythagorean triple (a, b, c) can be illustrated by a square and an L shape within another square. You are invited to find some triples for yourself.
Angle Trisection
Age 14 to 16 Challenge Level:
It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.
Why 24?
Age 14 to 16 Challenge Level:
Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
Pythagorean Triples I
Age 11 to 16
The first of two articles on Pythagorean Triples which asks how many right angled triangles can you find with the lengths of each side exactly a whole number measurement. Try it!
Pythagorean Triples II
Age 11 to 16
This is the second article on right-angled triangles whose edge lengths are whole numbers.
A Knight's Journey
Age 14 to 18
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
Yih or Luk Tsut K'i or Three Men's Morris
Age 11 to 18 Challenge Level:
Some puzzles requiring no knowledge of knot theory, just a careful inspection of the patterns. A glimpse of the classification of knots and a little about prime knots, crossing numbers and. . . .
Mediant Madness
Age 14 to 16 Challenge Level:
Kyle and his teacher disagree about his test score - who is right?
The Frieze Tree
Age 11 to 16
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Impossible Sandwiches
Age 11 to 18
In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.
Classifying Solids Using Angle Deficiency
Age 11 to 16 Challenge Level:
Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry
Proofs with Pictures
Age 14 to 18
Some diagrammatic 'proofs' of algebraic identities and inequalities.
The Pillar of Chios
Age 14 to 16 Challenge Level:
Semicircles are drawn on the sides of a rectangle. Prove that the sum of the areas of the four crescents is equal to the area of the rectangle.
Mouhefanggai
Age 14 to 16
Imagine two identical cylindrical pipes meeting at right angles and think about the shape of the space which belongs to both pipes. Early Chinese mathematicians call this shape the mouhefanggai.
A Chordingly
Age 11 to 14 Challenge Level:
Find the area of the annulus in terms of the length of the chord which is tangent to the inner circle.
Go Forth and Generalise
Age 11 to 14
Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
Pareq Exists
Age 14 to 16 Challenge Level:
Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.
Advent Calendar 2011 - Secondary
Age 11 to 18 Challenge Level:
Advent Calendar 2011 - a mathematical activity for each day during the run-up to Christmas.
NRICH is part of the family of activities in the Millennium Mathematics Project.