Math through Time

We start our journey through time and space around 3500 BC with the civilisation of the Sumer and later on the Babylonians, who developed an innovation to numeral systems. The Babylonian mathematical tablet Plimpton 322 that is believed to be written about 1800 BC. They introduced the positional notation. The value of every digit is determined by the symbol, as well as its position within the number. In our modern decimal system, also called Hindu–Arabic numeral system, the position of the digit determines whether it is multiplied by 1, 10, 100... The Babylonians used base 60 and made calculations more efficient, because huge numbers can be displayed with less digits. The power of the notational system allowed the development of geometric, arithmetic and algebraic methods that made very accurate calculations like an approximation of √2 possible. The Babylonian mathematical tablet YBC 7289 that is believed to be written about 1800-1600 BC. The diagonal shows an approximation of √2.
1 + 24/60 + 51/60² + 10/60³ ≈ 1.41421. The first known positional numeral system still influences us today: there are 60 minutes in an hour and 360 degrees in a circle.

The earliest fully developed numeral system with base ten The Egyptian numerals are based on multiples of ten without the usage of a pace-valued system. Instead multiples of these values were expressed by repeating the symbol as many times as needed. was used by the ancient Egyptians (circa. 3000 to circa 300 BCE) but unlike the Babylonian numerical system it was not positional. Evidence for Egyptian mathematics is limited to a few papyri, the largest one of them being the Papyrus Rhind. Left end of the front of the largest fragment of the Rhind Papyrus (now in the British Museum). Dating from about 1550 BCE, it covers various mathematical topics that we would now refer to as arithmetic, algebra and geometry in 84 problems. The problems (among other things) include fractional expressions, linear equations and the calculation of volumes and areas and stem from practical problems of trade and the market. The papyrus gives us insights on the mathematical methods of the time, one of them being a way of multiplication which requires only the ability to multiply by two and to add. It decomposes the smaller multiplicand into powers of two while creating a table of doublings of the second multiplicand.

Pythagoras of Samos (circa 570 - circa 495 BC) Bust of Pythagoras of Samos in the Capitoline Museums, Rome. was an ancient Greek philosopher, mathematician and founder of an influential religious-philosophical movement called Pythagoreanism. Though his work was not always based solely on his own ideas, the way he and his followers developed it by using axioms and logic to build a mathematical system was out of the ordinary and a significant legacy for his successors. While having made substantial contributions in other areas like astronomy and music, Pythagoras is most well-known for his mathematical discoveries particularly the Pythagorean theorem. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides (a² + b² = c²). There are several geometric and algebraic proofs of the theorem, interactively displayed is the Pythagorean proof, which is a proof by rearrangement. There is evidence that the Babylonians, Chinese and Egyptians knew of the mathematical relationship among the sides of a right triangle. The Babylonian tablet Plimpton 322 The Babylonian mathematical tablet Plimpton 322 that is believed to be written about 1800 BC contains a list of Pythagorean triples. records Pythagorean triples (three integers a, b, c such that a² + b² = c² ) and the Egyptians used knotted ropes to construct right angles. Evenly spaced knots, which can be formed into a 3,4,5 right triangle.

Hippasus of Metapontum (circa 530 – circa 450 BC) was a Pythagorean philosopher, who is sometimes credited with the discovery of irrational numbers. According to legend, the other Pythagoreans preached that all numbers could be expressed as a ratio of integers ( --> rational numbers) and drowned Hippasus for his proof of irrational numbers. Hippasos knew of the Pythagorean theorem from which resulted that a triangle with the length 1 for both short sides has a hypotenuse of length √2. He proved that √2 could not be expressed as a ratio of two integers This proof by contradiction appeared first as a full proof in Euclid's Elements, it is likely that Hippasus used a geometric proof. and therefore proved the existence of irrational numbers, which have an infinite amount of decimal places without terminating, nor ending with a repeating sequence. However, the evidence linking the discovery of irrational numbers to Hippasus and his death as a result of it, is confused.
On the right you can see the spiral of Theodorus, which is a spiral composed of right triangles, placed edge-to-edge. Theodorus of Cyrene, who was an ancient Greek who lived during the 5th century BC, proved the irrationality of all square roots of the non-square numbers up to 17.

Eratosthenes of Cyrene (circa 276 BC - circa 195) Eratosthenes teaching in Alexandria by Bernardo Strozzi (1635). was a Greek mathematician and astronomer. He is known for being the fist person to calculate the circumference of the earth, but he also proposed a simple algorithm for finding prime numbers: The Sieve of Eratosthenes.
First of all, all integers starting with two up to a freely selectable value are written down. The unmarked numbers are potential prime numbers. The smallest unmarked number is always prime, all multiples of that prime are marked as composite (not prime). The next larger integer is determined, which since it is not a multiple of numbers smaller than itself, is prime. All multiples are marked again. This process must be continued until the numbers are less than or equal to the square root of the bound.