May 30 – Happy Birthday, Georg von Peuerbach

Posted on May 30, 2014

This Austrian astronomer and mathematician—born on this date in 1493—accomplished a lot of things in his fields. Two of those accomplishments were creating better, more accurate eclipse tables and sine tables than had been available before.
Why were they better and more accurate?

Obviously, Peuerbach was careful with his computations. But in his sine table he also used what people call Arabic numbers, Hindu-Arabic numbers, or (these days) just numbers: 0123456789. He was one of the first mathematicians to promote the use of Arabic numbers in trigonometry.

Detour: What is a sine table?                                                            
Trigonometry is the math of right-angled triangles. Sine is the ratio of one side of a triangle to another side (since there are three sides to every triangle, there are other trigonometric ratios such as cosine and tangent).               
Back before there were electronic calculators, mathematicians made tables so that people using sine and other ratios wouldn't have to stop what they were doing and calculate a long, involved division problem—they could instead just consult the correct table.                 
For centuries people have used sine and cosine, and trigonometry in general, in surveying (measuring altitudes and distances) and in navigation. Nowadays they continue to use sine and cosine in space flight, analyzing sound waves and TV signal transmission, GPS and cell phones, and compressing digital information to make things like JPEG files.                                                                                           

Why are Hindu-Arabic numbers used so widely now?

Once upon a time there were many number systems, from Roman numerals and Japanese numerals to Mayan numerals and the Abjad number system. But now one set of numerals has spread all over the world. We still see some other numerals as well—such as Roman numerals labeling the volumes of a book or distinguishing one King Henry from another—but the 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?
This system was developed between the 1st and 4th Centuries by Indian mathematicians. Persian mathematicians adopted and developed the system, and the system was spread largely by the Arab civilization, which was the center of learning in the Western world from around the 8th Century to the 1200s. The Italian scholar Fibonacci encountered them in North Africa and helped to make them known throughout Europe, through his work.

By the mid-1500s these numbers were in common use in Europe, but it took early adopters like Peuerbach to make it happen!

Okay, so why this system? Why did the Hindu-Arabic system win out over all those other systems?

It has two great things: Positional place value, and Zero.

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

MCMLXIV – 1964

MMXVIII – 2018


(To learn more about Roman numerals, check out this earlier post.) 

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

And zero makes the whole place value system work, because if there are no “tens” in a number, you can plunk a zero in that spot and still make sense of the number. Without a zero as a placeholder, 105 and 15 would look like the same number!

So when you wish Georg von Peuerbach a happy birthday, also thank him for being an early adopter of our wonderful number system!

The Hindu-Arabic numerals did evolve and change from
ancient times to modern times.
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