Maths Assignment
Part A Q2-Maths Assignment 2012, Mrs Pillai Lvl 1
Irrational numbers are numbers that are neither whole numbers nor ratios of whole numbers. Irrational numbers are real numbers in the sense that they appear in measurements of geometric objects--for example, the number pi (II). However, irrational numbers cannot be represented as decimals, unlike rational numbers, which can be expressed either as finite decimals or as infinite decimals that eventually follow a repeating pattern. By contrast, irrational numbers have infinitely long decimal expansions that never form a repeating pattern. Thus, the number pi can never be written down exactly in decimal form, it can only be approximated, by decimals such as 3.14159. The golden ratio is another famous irrational number approximately equal to 1.618. It appears many times in geometry, art, architecture and other areas
Hippasus
Hippasus is credited with the discovery of irrational numbers. The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive and did not want their discoveries to get out. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. Along came Hippasus who discovered irrational numbers which consequently meant he was drowned. Hippasus was the disciple of Pythagoras; he was of course famous for his discovery of irrational numbers and more specifically his discovery of √2. There are many methods out there that prove the irrationality of √2. However this is the one that Hippasus used:
The proof of the irrationality of √2 is as follows:
1.Take a right triangle whose short sides are 1 unit in length
2.By the Pythagorean theorem, the diagonal is √2
3.Suppose that √2 is the ratio of two natural numbers,
Irrational numbers are numbers that are neither whole numbers nor ratios of whole numbers. Irrational numbers are real numbers in the sense that they appear in measurements of geometric objects--for example, the number pi (II). However, irrational numbers cannot be represented as decimals, unlike rational numbers, which can be expressed either as finite decimals or as infinite decimals that eventually follow a repeating pattern. By contrast, irrational numbers have infinitely long decimal expansions that never form a repeating pattern. Thus, the number pi can never be written down exactly in decimal form, it can only be approximated, by decimals such as 3.14159. The golden ratio is another famous irrational number approximately equal to 1.618. It appears many times in geometry, art, architecture and other areas
Hippasus
Hippasus is credited with the discovery of irrational numbers. The Pythagoreans were a strict society and all discoveries that happened had to be directly credited to them, not the individual responsible for the discovery. The Pythagoreans were very secretive and did not want their discoveries to get out. They all took oaths to ensure that their discoveries remained with the Pythagorean society. They considered whole numbers to be their rulers and that all quantities could be explained by whole numbers and their ratios. Along came Hippasus who discovered irrational numbers which consequently meant he was drowned. Hippasus was the disciple of Pythagoras; he was of course famous for his discovery of irrational numbers and more specifically his discovery of √2. There are many methods out there that prove the irrationality of √2. However this is the one that Hippasus used:
The proof of the irrationality of √2 is as follows:
1.Take a right triangle whose short sides are 1 unit in length
2.By the Pythagorean theorem, the diagonal is √2
3.Suppose that √2 is the ratio of two natural numbers,