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 v2. There are many methods out there that prove the irrationality of v2. However this is the one that Hippasus used:The proof of the irrationality of v2 is as follows: 1. Take a right triangle whose short sides are 1 unit in length 2.
By the Pythagorean theorem, the diagonal is v2 3. Suppose that v2 is the ratio of two natural numbers, v2=m/n 4. Suppose that m/n has been reduced to its lowest common form by division 5. It follows that either m and n are both odd, or that m is odd and n is even, or that m is even and n is odd (if not, we could reduce m/n even further by dividing both numbers by 2) 6.
Square both sides of v2=m/n, so that 2=m2/n2 7.Then 2n2=m2, so that m2 is even, and therefore m is even 8. If m is even, then m=2x, where x is some other natural number 9. Squaring this, it follows that m2=4x2=2n2 10. It follows that n2=2x2, and therefore n2 is even, which means that n, being a natural number, must be even 11.
So we've reached a contradiction: although we assumed that m and n cannot both be even, it now turns out they both are. It therefore follows that v2 cannot be expressed as the ratio of two natural numbers, and must therefore be in another class of numbers EulerLeonhard Euler was born on 15 April 1707 in Basel, Switzerland and died on 18 Sept 1783 in St Petersburg, Russia. Leonhard Euler is most famous for his discover of Euler’s number which funnily enough is named after him. This number can never be exact, it is an irrational number which goes on and on.
The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2. 718055556. This number is more commonly than not referred to as “e”. The number e is a famous irrational number, and is one of the most important numbers in mathematics.
e is the base of the Natural Logarithms.On the other hand Common Logarithms have 10 as their base. There are many ways of calculating the value of e, but none of them ever give an exact answer, because e is irrational (not the ratio of two integers). For example, the value of n (1 + 1/n)n approaches e as n gets bigger and bigger: n (1 + 1/n)n 1 2. 00000 2 2. 25000 5 2.
48832 10 2. 59374 100 2. 70481 1,000 2. 71692 10,000 2. 71815 100,000 2. 71827 From this we can see that the increase in the number get steadily lower as n increases.
So to get e exactly we obviously know that you must have a very large number.Many mathematician have tried to find out the exact number e but as this number goes on and on this is seemingly impossible. These are the first few digits of e : 2. 718281828459045235360287471352662497757247093699959574966967 627724076630353 e is commonly used in exponential graphs and natural logarithms. Exponential graphs are graphs that have a rapid increase at first then a slow/steady increase as the graph goes on.
Just like the graph above. Common Logarithms have 10 as their base while a natural logarithm has e as its base. Natural Logarithms are used mainly in calculus. Ahmed Ugool
