Fractional calculus is a generalisation of ordinary calculus where you can differentiate n times when n is not a whole number.
Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.
Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”
First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...
You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.
The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?
Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?
Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
Generalise the sum of a GP by using derivatives to make the coefficients into powers of the natural numbers.
For which values of n is the Fibonacci number fn even? Which Fibonnaci numbers are divisible by 3?
Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information.
Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?
Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
Can you use the diagram to prove the AM-GM inequality?
This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.
What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.
What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?
Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .
Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?
It would be nice to have a strategy for disentangling any tangled ropes...
An account of some magic squares and their properties and and how to construct them for yourself.
Generalise this inequality involving integrals.
Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?
Given a set of points (x,y) with distinct x values, find a polynomial that goes through all of them, then prove some results about the existence and uniqueness of these polynomials.
Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?
Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?
Can you see how to build a harmonic triangle? Can you work out the next two rows?
Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
Can you explain the surprising results Jo found when she calculated the difference between square numbers?
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?
Can you find the values at the vertices when you know the values on the edges?
Can you tangle yourself up and reach any fraction?
When if ever do you get the right answer if you add two fractions by adding the numerators and adding the denominators?
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
An article which gives an account of some properties of magic squares.
If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?
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.
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
Can you find the area of a parallelogram defined by two vectors?
What is the total number of squares that can be made on a 5 by 5 geoboard?
Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?
Build gnomons that are related to the Fibonacci sequence and try to explain why this is possible.
These gnomons appear to have more than a passing connection with the Fibonacci sequence. This problem ask you to investigate some of these connections.