Making sense of irrational numbers - Ganesh Pai
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Like many heroes of Greek myths, the philosopher Hippasus was rumored to have been mortally punished by the gods. But what was his crime? Did he murder guests or disrupt a sacred ritual? No, Hippasus's transgression was mathematically proving the hitherto unprovable. Ganesh Pai describes the history and math behind irrational numbers.
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Irrational numbers form a subset of real numbers. To learn more, watch what are real numbers? There are many complicated explanations of what real numbers are, but the simplest explanation is that they are any number that can be represented on a number line. If we look one step further, these numbers can be broadly classified as rational and irrational numbers. Rational numbers are those that can be represented as a ratio of two integers with no common factor. Irrational numbers, on the other hand, cannot be expressed as a ratio of two integers. When expressed in decimal notation, irrational numbers are non-terminating non-recurring decimals.
So, what exactly do we mean by non-terminating non-recurring? This video might just give you the answer. If we calculate the square root of 2, we get a value like 1.4142135… This value is ‘non-terminating’ as you will not find any repetitive patterns in the digits after the decimal. Therefore, we call this a ‘non-recurring’ decimal. For a deeper understanding of irrational numbers and operations on them, visit this link.
Pi is the most commonly known ‘special irrational number’. Watch this TED-Ed lesson: The infinite life of pi by Reynaldo Lopes for more. However, there are many other ‘special irrational numbers’ such as e and the golden ratio. Read: The other irrational numbers we could celebrate instead of pi to find out more about these numbers.
These numbers are also referred to as transcendental numbers. Basically, they are non-algebraic numbers, numbers that are not roots of any algebraic equation with rational coefficients. For more on transcendental numbers, check out The 15 Most Famous Transcendental Numbers and Transcendental Numbers by Numberphile.
In mathematics, irrational numbers are also referred to as incommensurable numbers. The story of Hippasus highlights the importance of being able to change our minds when presented with evidence that disproves our beliefs. P. C Hodgell said: “That which can be destroyed by the truth should be.” More on the concept of ‘philosophical incommensurability’ can be found in Eliezer Yudowsky’s essay on the 12 Virtues of Rationality.
How can we compare ‘mathematical incommensurability’ with ‘philosophical incommensurability’? This article gives a good insight into how these can be compared, Take a look to find out more!
Finally, just for fun you can try this to find where your date of birth lies in the decimal part of Pi.
So, what exactly do we mean by non-terminating non-recurring? This video might just give you the answer. If we calculate the square root of 2, we get a value like 1.4142135… This value is ‘non-terminating’ as you will not find any repetitive patterns in the digits after the decimal. Therefore, we call this a ‘non-recurring’ decimal. For a deeper understanding of irrational numbers and operations on them, visit this link.
Pi is the most commonly known ‘special irrational number’. Watch this TED-Ed lesson: The infinite life of pi by Reynaldo Lopes for more. However, there are many other ‘special irrational numbers’ such as e and the golden ratio. Read: The other irrational numbers we could celebrate instead of pi to find out more about these numbers.
These numbers are also referred to as transcendental numbers. Basically, they are non-algebraic numbers, numbers that are not roots of any algebraic equation with rational coefficients. For more on transcendental numbers, check out The 15 Most Famous Transcendental Numbers and Transcendental Numbers by Numberphile.
In mathematics, irrational numbers are also referred to as incommensurable numbers. The story of Hippasus highlights the importance of being able to change our minds when presented with evidence that disproves our beliefs. P. C Hodgell said: “That which can be destroyed by the truth should be.” More on the concept of ‘philosophical incommensurability’ can be found in Eliezer Yudowsky’s essay on the 12 Virtues of Rationality.
How can we compare ‘mathematical incommensurability’ with ‘philosophical incommensurability’? This article gives a good insight into how these can be compared, Take a look to find out more!
Finally, just for fun you can try this to find where your date of birth lies in the decimal part of Pi.
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