Archimedes of Syracuse, in Sicily, (287-212 BCE) devices the method of approximating the area of the unit circle using inscribed and circumscribed polygons, now known as the classical method of exhaustion. The area of a regular polygon of n sides inscribed in a unit circle is smaller than the area of the circle. As n approaches infinity, the polygonal areas should approach the area of the unit circle.

*move the slider to see the change in the number of sides of polygon and area of regular polygon.

The Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. The following applet shows the concept of upper and lower sum of Riemann integral which calculates the area of rectangles.

Move the slider to increase the the number of the partitions and see how the upper sum and lower sum approach each other as well as to the original integral value.

Definite integrals are integrals with certain known limits. Here you can change the integral function and also the limits and view the geometric interpretation of integral which is area under the curve. Also you can manually change the positions of limit points A and B.

The following applet shows a function and a tangent to that curve at some point and also slope of that tangent.

Here you can change the function in the input box. Also you can slide point A along X-axis to see tangent at various points on that curve.

In this applet you can input the desired function. The applet will plot the function and its derivative on the geometric view. Also you can slide the point on X-axis.

For the following applet enter your function in the input box and move the point on the X-axis which will trace out your function.

The following applet shows a complex number and its geometric interpretation. Either you can enter a complex number in the input box or you can slide z over the complex plane.

The following applet shows addition of two complex numbers and its geometrical view. Either you can enter complex numbers in the input box or you can slide complex numbers over the complex plane.

The following applet shows subtraction of two complex numbers and its geometrical view. Either you can enter complex numbers in the input box or you can slide complex numbers over the complex plane.

The following applet shows multip of two complex numbers and its geometrical view. Either you can enter complex numbers in the input box or you can slide complex numbers over the complex plane.

The following applet plots the all complex nth roots of complex numbers on the complex plane up to n=15. You can increase the slider to see all nth roots which are all on the circle and equi-spaced.

There was a young and adventurous man who found among his great-grandfather’s papers a piece of parchment that revealed the location of a hidden treasure. The instructions read:

“Sail to __ north latitude and __ west longitude where you will find a deserted island. (Actual coordinates have been omitted so as not to give away the secret and cause a gold rush among readers.) There is a large meadow on the north shore of the island where there are two huge stones. Also there is a big pine tree at some distance. Start from the pine tree and walk to the one stone counting the steps. At the stone you must turn right by a right angle and take exactly same number of steps as many steps as you just took to reach the stone. Put here a spike into the ground. Now return to the pine tree and walk to the other stone counting the steps. At the second stone you must turn left by a right angle and again take exactly same number of steps as many steps as you took to reach the second stone. Put there another spike into the ground. Dig halfway between the spikes; the treasure is there.”

The young man charted a ship and sailed to the South Seas, where he found the island. But to his great sorrow, the pine tree was gone. Too much time had passed since the document had been written; the elements had disintegrated the wood and returned it to the soil, leaving no trace even where it once had stood. The young man, in an angry frenzy, began to dig at random, but the island was too big. So he sailed back empty-handed. The treasure is probably still there.

A sad story, but what is sadder is the fact that the fellow might have had the treasure, if only he had known a bit about mathematics, and specifically the use of imaginary numbers. Can you find the treasure for him?

We have worked everything said in the riddle in the applet below. Try the situation by changing the position of the pine tree yourself .

The Pythagorean Tree is named after the Greek mathematician Pythagorus because the construction demonstrates the geometric proof of the Pythagorean Theorem that the sum of the areas of the squares along the two sides of a right triangle is equal to the area of the square along the hypotenuse. The Pythagorean Tree was apparently first drawn by Albert E. Bosman (1891-1961) around 1942. Bosman was a Dutch electrical engineering and mathematics teacher.

*Move the BLUE point to see different shapes.