Posted
on May 30, 2014
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? |
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
MDCCLXXVI – 1776
(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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