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Johann Carl Friedrich Gauss ( 1777 –1855). Gauss ’ s father owned a butcher shop. The first sign of his son ’ s unusual gifts occurred when he was three. His father was adding up a column of numbers and made a mistake; young Carl corrected him.
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Gauss’s father owned a butcher shop. The first sign of his son’s unusual gifts occurred when he was three. His father was adding up a column of numbers and made a mistake; young Carl corrected him. Later on, when he was in school, his class was making too much noise and the teacher punished them by having them all add up the numbers from 1 to 100. Thinking that this would take them quite some time, he was startled when Gauss quickly wrote the correct answer (5050) on his slate. There is some doubt that either of these stories actually happened . . . .
But let us consider the problem of adding up the numbers from 1 to n, for any fixed number n. Here are the first few examples: 1 = 1 1 + 2 = 3 1 + 2 + 3 = 6 1 + 2 + 3 + 4 = 10 ... By staring at this for a while, we might be able to guess the answer: the sum of the numbers from 1 to n is It is not difficult to check that it works for the examples above (and others). But that is not a proof that it is always true. For that, we use what is called “proof by induction".
1 + 2 + 3 + . . . + n = First observe that the result is true when n = 1. Then suppose it is known for some fixed value of n. We want to prove that it must then be true for n + 1. To say it holds for n is to say that 1+2+3+4+…+n = We use this to see that 1 + 2 + 3 + 4 + … + n+ (n + 1) = + (n + 1).
Taking out a common factor of (n + 1), we see that the right side is + (n+1) = Rearranging this last formula gives which is the required formula for the sum of the numbers up to (n + 1). This concludes the proof by induction. The result holds for n = 1, and if it is true for any n, then it is also true for (n + 1). This means it must hold for any n.
Another way to look at it is that if m is the smallest positive integer for which the formula fails, then it holds for (m - 1). But then the inductive step shows that it is true for m, a contradiction.
The Pythagorean Theorem Pythagoras was a Greek who lived in Sicily. His famous result concerns a right angled triangle. If the sides have lengths a, b, c, with c the length of the hypotenuse, then the Pythagorean Theorem says that a2 + b2 = c2. c b a
Here is a pictorial version. The squares have areas a2, b2, and c2, and the theorem says the area of the large one is the sum of the areas of the other two: ere c b a a2 + b2 = c2
This picture shows how the two smaller squares can be cut up and reassembled to make the big square. It constitutes a proof of the theorem. However, to make it into a proper proof, it is necessary to convince yourself that all the pieces fit together as shown. It helps to notice that each side of any of the pieces is parallel to a side of one of the three squares.
Just to suggest that it is more subtle than it appears, this picture shows how it works in the special case in which the two shorter sides are equal, i.e., when the triangle is isosceles.
Uncountability of the Real Numbers We have seen that is not a rational number, so there definitely are numbers on the real line which are not rational. Now we show that there are so many of them that they are not countable. Note that each real number can be written as a decimal, like 0.345234 . . . or 324.877456372 . . . Unfortunately, there is a small ambiguity: the number 0.999999 . . . (with 9 repeated indefinitely) is the same as the number 1.0. Similarly, any number which “terminates"', i.e., which has only finitely many nonzero digits, ending in an infinite string of zeroes, can be written instead as a number that ends in a string of 9’s.
To resolve the ambiguity, let us agree always to write such numbers with an infinite string of zeroes rather than 9’s. Now suppose it were possible to enumerate the real numbers, say as r1, r2, r3, . . . . Let us construct a new real number x by describing its decimal expansion. The number x will begin with “0.”. Then for the first decimal place, we will choose a digit which is not 9 and not equal to the first decimal place of r1. For the second decimal place, we will choose a digit which is not 9 and not equal to the second decimal place of r2. For the third decimal place, we will choose a digit which is not 9 and not equal to the third decimal place of r3.
For each n, we will choose for the nth decimal place of x a digit which is not 9 and not equal to the nth decimal place of rn. This completely describes x. It is a legitimate decimal number between 0 and 1. But it cannot equal any of the numbers r1, r2, r3, . . . . This is because it differs from each of them in at least one decimal place. This is a contradiction, since every real number is assumed to appear in the list. The contradiction proves that it was wrong to assume that the real numbers are countable, so they must be uncountable. QED
The Bridges of Königsberg (Kaliningrad)
Is it possible to devise a path which crosses each bridge once and not more than once?
Leonhard Paul Euler (1707 –1783), Swiss mathematician and physicist
L Leonhard Euler in 1735
Euler extracted the essential points about the problem of the bridges in Königsberg and expressed them in a graph which was easier to understand. • The mathematicians working on the Trans-Siberian Railway problem thought they had also isolated the essential elements, but they missed an essential feature of the situation.
Consider a population of rabbits. For simplicity, let us keep track of the females and ignore the males. Let’s assume that females do not reproduce in their first year, but after that, they produce one additional female each year. We shall also ignore deaths.
