One day the nuns asked the students in Karl Fredrich Gauss' class to add up all the numbers from 1 to 100. Gauss imagined the sequence written twice --- one row beneath the other and in reverse order. Each pair --- like 1 and 100, 2 and 99, and so on --- would add up to 101, and there were 100 such pairs. So the sum can be calculated as 1/2 (first + last)(number of terms) = 1/2(101)(100) = 5,050. The 1/2 is because the sequence is written twice.
When he was just 19, Isaac Newton developed the binomial theorem. This provides a simple way to calculate powers of binomial expressions --- things that look like (a + b)^n. The theorem says that (a + b)^n = [n 0]a^nb^0 + [n 1]a^(n-1)b^1 + ... +[n (n-1)]a^1b^(n-1) + [n n]a^0b^n, where " ..." means "continue on like this" and [n k] means "the number of different ways that you can choose k things from a set of n things." In more mathematical notation, [n k] = n!/k!(n-k)! and z! = 1 X 2 X 3 X ... X z. For example, (a + b)^3 = a^4 + 4a^3b+ 6a^2b^2 + 4ab^3 + b^4.
The Pythagorean theorem describes the relationship between the lengths of the sides of a right triangle. Across from the right angle is the hypotenuse, the longest side of the triangle. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. One possibility is 3, 4 and 5 --- the hypotenuse is length 5. 5^2 = 3^2 + 4^2 because 25 = 9 + 16. The relationship works both ways --- if the ratios of the lengths are 3, 4 and 5, the largest angle in the triangle is exactly 90 degrees. Egyptians had a device with three wooden stakes connected with a ropes of lengths 3, 4 and 5 that they used to redraw the corners of property lines after the annual flooding of the Nile.
Rene Descartes habitually stayed in bed all morning, writing and working on mathematics. One day he was watching a fly walking across the ceiling and thought that if there were numbers along the edges of the ceiling like along a ruler, it would be easy to describe the fly's path as a sequence of numbers. From this basic idea came what is now known as analytic geometry, a system for turning mathematical formulas into pictures. The marriage of algebra and geometry has turned out to be a trick of enormous value, making possible the invention of calculus.