Once Upon a Math Quiz

Answer: 60

The Babylonian number system was of a base-60 design. These were found on many stone tablets used for mathematical calculations. Sixty, a large number, was further broken down into ones (vertical wedges) and tens (corner wedges). Therefore, a number like 43 would have been written as four corner wedges and three vertical wedges. Once you got to sixty, there would be a new column for numbers. So, a large number like 135 would be written as two vertical wedges in the 60s column (120) and then one corner wedge and five vertical wedges in the ones column (15). Add them together and you get 135.

Now, why is this significant? Well, when you have a number like 60, it has many divisors. When you're calculating fractions, when you have more divisors, it makes it easier to write down simple fractions. This base-60 system is still around today. When you tell time, you're using the base-60 system. When you're calculating degrees in a circle, there it is again!

Answer: True

An ancient Babylonian tablet (YBC 7289) found an approximation for the square root of two. The Babylonians had it as 1;24,51,10 (which, in decimals, is approximately 1.4146296286) and squared, it comes very very close to 2. On the same tablet was posed a mathematical question where they were calculating the diagonal of a square, so, they knew the formula for the Pythagorean Theorem for an isosceles triangle, at least.

This tablet was carbon-dated to predate Pythagoras by about 1200 years!

Answer: Simple interest

The Babylonians used the exponential function to calculate the value of a loan to be repaid after certain lengths of time. They listed problems that would ask about a certain amount of interest and sought to calculate when the value of the loan would double. So, the exponential function as well as exponential growth was well known to the Babylonians.

Answer: Parallel lines never meet

When Euclid went about proving things geometrically in "Elements", he stated five postulates that formed the basis for his geometry. These postulates were axioms, as in, unprovable truths, but they hold true for all of his geometric constructs. They were, in order:
1. Two points can be connected by a single straight line segment.
2. Line segments can be extended to infinity.
3. Circles can be drawn using a line segment as a radius and fixing one of the ends as the center of the circle.
4. All right angles are equal to each other.
5. Parallel lines never meet.

Non-Euclidean geometry is geometry on curved surfaces. In this case, postulate 5 is negated (look at longitudinal lines, for example, at the equator they're parallel but they converge at the poles).

Answer: Pentagon

A dodecahedron is a solid made up of twelve faces, each of them being a regular pentagon. It is one of the five Platonic solids. The pentagon, in particular, was difficult to draw with only a straight edge and a compass because the golden ratio, e, is an irrational number and you cannot draw a line segment of length e precisely. Similarly, 72 degrees, as an inner angle, was difficult to measure out, until Euclid provided the proof for how it's done. Euclid painstakingly went through steps in order to construct the pentagon inscribed in a circle (actually, a decagon first, as it is easier, and then a pentagon) and was the first known person to write of the existence of the golden ratio.

Answer: Primes

In "Elements" book IX, proposition 20, Euclid wrote down a simple proof showing that there are an infinite amount of prime numbers. His proof is still taught today and is easy to understand and re-prove. It is a proof by contradiction:
Say there are a finite number of primes, let's call that number N.
If we multiply all the primes and then add 1, we get a number, let's call it P.
p_1*p_2*p_3...*p_N + 1 = P
Now, either P is prime, because none of the previous primes is a factor of it (when you divide P by any other prime you get a remainder of 1) or P s a composite number. However, P must have a divisor, p', that is not part of the original set because all the other primes are not factors of P. Therefore, another prime exists, so, the set of prime numbers cannot be finite.

Answer: Pythagorean

The Pythagoreans were to have developed a very simple proof that the square root of 2 is irrational. It is often attributed to Hippasus of Metapontum. Ironically, the Pythagoreans murdered Hippasus for writing this proof before embracing it (but only geometrically).

Aristotle refers to this proof and sketched it out in his "Analytica Priora". Though it is a very simple proof, and introduces the existence of the irrational numbers, the Greeks themselves did not give this much thought and would continue their studies of the rational numbers. The proof itself is quite simple:
Suppose the square root of 2 is rational, therefore, it can be expressed in a fraction of lowest terms where a/b have no common divisors. Therefore, a^2/b^2 = 2.
If we transfer sides, then a^2 = 2*b^2. Since a and b have no common divisors, then, since 2 is a divisor of a^2, then b must be odd.
Since 2 is a divisor of a^2, then a must be even, so, a = 2*p (if a was odd, then a^2 must be odd), we can go and substitute that into the equation above:
(2*p)^2 = 2*b^2
Therefore, 4*p^2 = 2*b^2.
This can be reduced to 2*p^2 = b^2. This, as above shown for a, shows that b is even, contradicting the statement above that b was odd.
Therefore, no rational fraction, a/b exists for the square root of 2. So, it must be irrational.

Answer: Archimedes

Archimedes, perhaps one of the more prominent mathematicians of his time, and one of the few where biographical information survived to modern times, wrote down many mathematical proofs. In fact, Plutarch wrote of Archimedes' work that:

"It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations."

When comparing a triangle to a circle, Archimedes discovered that "the area of a circle with a radius of r is equal to that of a triangle whose base is equal to the circumference C of the circle and whose height is r." This gives us the equation:

A = (1/2)*r*C

This shows us that there is a set ratio that relates the square of the radius to the area of the circle, and that ratio is pi, which is sometimes called Archimedes' Constant.

Answer: Parabola

Archimedes wrote "The Method of Mechanical Theorems" which was believed to have been almost exclusively work derived by Archimedes himself. While Euclid's "Elements" is likely to have been a compendium of previous mathematicians' work, "The Method" seems to be an original piece.

In it, Archimedes inscribes a regular heptagon with a straight edge and a compass, trisects an angle with those same tools, and provides a proof for "the area of a parabola to be 4/3 the area of the inscribes triangle of the greatest area".

In fact, Archimedes is solving, in terms of modern calculus, the integral from 0 to 1 of x^2 dx = 1/3 and using that same method to calculate other sections of the parabola. Genius!

Answer: Real numbers

Eudoxos of Cnidos developed a theory that, in 1872, provided Richard Dedekind the basis for the modern theory of real numbers, and the construction of all numbers, including irrational ones. He stated, in a precursor to the epsilon-delta proofs, that where numbers are commensurable, even if they do not really exist (irrational), then one can find rational proportions between them.

The ancient Greeks hated irrational numbers. They knew that they existed, but accepted them only geometrically, because then they had a purpose. Numerically, they didn't make sense to the Greeks, and they shied away from acknowledging them in mathematics, and others followed suit, for hundreds of years.