May 30 – Happy Birthday, Georg von Peuerbach

nPostednon May 30, 2014

nThisnAustrian astronomer and mathematician—born on this date inn1493—accomplished a lot of things in his fields. Two of thosenaccomplishments were creating better, more accurate eclipse tablesnand sine tables than had been available before.

nWhynwere they better and more accurate?

nObviously,nPeuerbach was careful with his computations. But in his sine table henalso used what people call Arabic numbers, Hindu-Arabic numbers, orn(these days) just numbers: 0123456789. He was one of the firstnmathematicians to promote the use of Arabic numbers in trigonometry.

nDetour:nWhat is a sine table?

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Trigonometrynis the math of right-angled triangles. Sine is the ratio of one sidenof a triangle to another side (since there are three sides to everyntriangle, there are other trigonometric ratios such as cosine andntangent).

nBacknbefore there were electronic calculators, mathematicians made tablesnso that people using sine and other ratios wouldn’t have to stop whatnthey were doing and calculate a long, involved division problem—theyncould instead just consult the correct table.

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nForncenturies people have used sine and cosine, and trigonometry inngeneral, in surveying (measuring altitudes and distances) and innnavigation. Nowadays they continue to use sine and cosine in spacenflight, analyzing sound waves and TV signal transmission, GPS andncell phones, and compressing digital information to make things likenJPEG files.

nWhynare Hindu-Arabic numbers used so widely now?

Oncenupon a time there were many number systems, from Roman numerals andnJapanese numerals to Mayan numerals and the Abjad number system. Butnnow one set of numerals has spread all over the world. We still seensome other numerals as well—such as Roman numerals labeling thenvolumes of a book or distinguishing one King Henry from another—butnthe familiar 0123456789 has become the dominant system.

This telephone number pad
is from Egypt. The numerals
on the left are the Western
“Arabic numerals,” and those on
the right are the Eastern “Arabic
numerals.” Confusing, huh?

nThisnsystem was developed between the 1^st and 4^thnCenturies by Indian mathematicians. Persian mathematicians adoptednand developed the system, and the system was spread largely by thenArab civilization, which was the center of learning in the Westernnworld from around the 8^th Century to the 1200s. ThenItalian scholar Fibonacci encountered them in North Africa and helpednto make them known throughout Europe, through his work. n

nBynthe mid-1500s these numbers were in common use in Europe, but it tooknearly adopters like Peuerbach to make it happen!

nOkay,nso why this system? Why did the Hindu-Arabic system win out over allnthose other systems?

nItnhas two great things: Positional place value, and Zero.

nThenHindu-Arabic system is not the ONLY system to ever have these greatnthings, but it is one of the first to combine both. If you have evernseen a comparison of dates in Roman numerals and our (Arabic)nnumerals, you know that ours can be a space saver:

n MCMLXIVn– 1964

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n MMXVIIIn– 2018

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n MDCCLXXVIn– 1776

n(Tonlearn more about Roman numerals, check out this earlier post.)

nOursnis a space saver because it has “positional” place value. Annumber 2 in the leftmost spot of a whole number is exactly that – 2n– but when it is in the second spot from the left, it represents 2ntens (20), and when it is one more spot away from the left, itnrepresents 2 hundreds (200), and so forth. n

nAndnzero makes the whole place value system work, because if there are non“tens” in a number, you can plunk a zero in that spot and stillnmake sense of the number. Without a zero as a placeholder, 105 and 15nwould look like the same number!

nSonwhen you wish Georg von Peuerbach a happy birthday, also thank himnfor being an early adopter of our wonderful number system!