![]() ![]() In the case that A = 1729, we have 2 possible ways of finding distinct integer solutions: The general problem referenced above is finding integer solutions to the below equation for given values of A: Ramanujan was profoundly interested in number theory – the study of integers and patterns inherent within them. Ramanujan remarked in reply, ” No Hardy, it’s a very interesting number! It’s the smallest number expressible as the sum of 2 cubes in 2 different ways!” Visiting him in hospital, Hardy remarked that the taxi that had brought him to the hospital had a very “rather dull number” – number 1729. H Hardy led him to being invited to study in England, though whilst there he fell sick. His correspondence with the renowned mathematician G. The Indian mathematician Ramanujan (picture cite: Wikipedia) is renowned as one of great self-taught mathematical prodigies. Ramanujan’s Taxi Cabs and the Sum of 2 Cubes Sometimes Pi is approximated to 3.141592653589, bringing us to the second 9 in the string.If you are a teacher then please also visit my new site: for over 2000+ pdf pages of resources for teaching IB maths! Every decimal we add makes the number more exact. The simplest approximation of Pi used is typically something like 3.14 or 3.1415, although lots of examples use 3.14159. We can never write Pi as an exact string of numbers (as the string never ends), so we will always have to approximate. So we can, for instance, find the circumference of a circle by only knowing Pi and the diameter of the circle. Pi is useful because it is an easy to use mathematical constant that can calculate a circle (to varying degrees of accuracy based on how many digits are used). The result of Pi is always the same, approximately 3.14159… Pi (π) a mathematical constant, that represents the ratio of a circle’s circumference (C) to its diameter (d) expressed as π = C/ d. With that in mind, this page looks at “ the last digit of Pi” and this page asks “ Is Pi Infinite?“. This is just a technicality, but Pi shouldn’t be considered “never ending” or “infinite”, it is simply irrational, meaning that you can’t get a natural number by multiplying Pi by some other natural number. Never ending is not the same as irrational as explained here. Pi is irrational, as it doesn’t repeat a pattern, but that doesn’t mean we should say “Pi never ends” (for much the same reasons we wouldn’t say Pi is an infinite number). Since infinity is a concept, and not a number, neither rational or irrational numbers are “infinite numbers”. Generally, there are many different types of infinity. Both rational and irrational numbers are infinite in that they “can’t be expressed by a real number”, but only irrational numbers “can’t be expressed as a fraction and don’t repeat a pattern”. 333… (a rational number, a repeating pattern, with an infinite number of digits). This is different from something like 1/3 which can be expressed as. Given the aforementioned, Pi is not an “infinite number” that “never ends”, it is a real number between 3 and 4, with a non-repeating pattern that cannot be fully expressed as a ratio of integers (i.e. as a fraction). ![]() However, technically speaking, infinity is an abstract concept and not a number. Pi’s decimal representation never settles into a permanent repeating pattern and can’t be fully expressed on paper, so it is infinite in these ways. Irrational refers to a real number that “can’t be expressed as a fraction and doesn’t repeat a pattern”.Infinite is a concept that means “can’t be expressed by a real number”.Pi is not an infinite number, it is an irrational number. Proof that Pi is irrational can be found here. Proof that Pi is Irrational: Johann Heinrich Lambert proved that Pi was irrational in 1761. ![]() A SciShow video on mathematics‘ most delicious irrational number π (although the golden ratio φ also looks rather delicious). ![]()
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