It's interesting that throughout history, people have yelled "Impossible!" when some new type of number was proposed. It was one hundred years later when the Greek astronomer Eudoxus (around 370 B.C.) concluded that because we can measure irrational distances (as we did above), then irrational numbers must exist.
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He continued on to √16 = 4, constructed one more, √17, then stopped.Īnd so now you know how to construct the square root of any number using a straight edge, a pencil and a set-square. He kept going and found that the next one to have a "rational" length was √9 = 3. Theodorus had discovered one hypotenuse with a rational number length. This gives us the length √3 after we apply Pythagoras' Theorem to the new triangle.ĭo it again, and you now get the length √4 = 2. Then, extend a line with length 1 unit (using your 1-unit measuring stick) at right angles to the first hypotenuse as follows. Start with a right triangle with equal sides 1, giving a hypotenuse of √2 (which of course was a problem, because this distance didn't officially exist): (Unfortunately we no longer have the proofs.) He also went on to construct these supposedly non-existent distances. He apparently proved that the square roots of 2, 3, 5, 6 and so on up to 17 were all irrational, except the perfect squares 4, 9, 16. (Cyrene is now called Shahhat, in Libya.) mathematician and was born around 100 years after Pythagoras. Theodorus of Cyrene was a 5th century B.C. How could they measure these distances if they didn't actually exist? Theodorus of Cyrene The length of the hypotenuse involved a square root:ĭepending on the values of a and b, we could easily get irrational values for c. However, since they believed irrational numbers did not exist there was a problem when they extended the Pythagorean formula to other values. Each number is an integer and for the ancient Greek mathematicians, this presented no problem. In my diagram above, we have the common 3-4-5 triangle. You normally see Pythagoras' Theorem written as follows, where c is the hypotenuse and a and b are the lengths of the other 2 sides: Pythagoras (or someone in his metaphysical school of mathematicians) had shown the famous result that for a right angled triangle, the area of the square on the hypotenuse (in green below) equals the sum of the areas of the squares on the other 2 sides (the 2 light red squares). So they only accepted rational numbers and any other numbers were "unmeasureable". They believed the only meaningful numbers were the natural numbers (1,2,3.) and any ratios involving these numbers (like 5/2, 7/9, etc). In the 5th century B.C., mathematicians were fascinated - yet exasperated - with irrational numbers. First, some background on several very important ancient mathematicians and how irrational numbers came to be. The image below explains the relationship between Irrational numbers and Real Numbers.Let's start with an interesting question.Ĭan you construct the length √7 just by using a measuring stick, a pencil and a set-square? (A set-square is a triangular device used for producing right angles.)īefore we answer this question. For example, √2, √3, and π are all irrational numbers and can’t be written as fractions. Although, irrational numbers can be expressed in the form of non-terminating and non-recurring fractions. The differences between rational and irrational numbers can be learned here. But irrational numbers are different from rational numbers as they can’t be written in the form of fractions. Irrational numbers come under real numbers, i.e. Golden ratio “ϕ” = This is an irrational number and its application is found in many fields like computer science, design, art, and architecture.In this case, also, people tried calculating it up to a lot of decimals but still, no pattern was found. Euler’s number “e” = Euler number is also very popular in mathematics.People have calculated its value up to quadrillion decimal but still haven’t found any pattern yet. √2, √3, π, e are some examples of irrational numbers. ISRO CS Syllabus for Scientist/Engineer Exam.ISRO CS Original Papers and Official Keys.GATE CS Original Papers and Official Keys.DevOps Engineering - Planning to Production.Python Backend Development with Django(Live).
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