Question: When did exp onents come into existence? Why is a
number to the zero
power always equal one?
Findings:
When did exponents come into existence?
• Nicole Oresme (1323-1382) used powering, but did not use raised
numbers.
• Nicolas Chuquet (1445?-1500?) used raised numbers, however in his
work, 123 actually meant 12x3
• Pierre Hérigone (1580-1643) wrote a, a2, a3, etc. (vol. 1, page 202, and
Ball).
• James Hume used Roman numerals as exponenets (vol. 1, pages 345-346)
An example of that is as follows: he wrote Aiii for A3
• Rene Descartes (1596-1650), in 1637, used the modern notation for
exponents. They were only positive at this point.
• Nicolas Chuquet used negative integers as exponents in 1484.
• Isaac Newton was the first to use negative exponents in June 1676 in a letter
to
Henry Oldenburg. Henry Oldenburg was the secretary of the Royal Society.
Newton described his discovery of the general binomial theorem in his letter.
(Cajori 1919, page 178).
• It is said that John Wallis was the first to suggest the use of negative
exponents,
but did nothing to use negative numbers in this way.
• Nicole Oresme (1323-1382) was the first to use fractions as exponents . Her
specific notation remained unnoticed for many years. An example of her notation
is as follows: she wrote
to re present
.
• Isaac Newton used fractional exponents in the notation we currently know of.
Why is a number to the zero power always equal one?
• After looking at several examples with varying exponents, the
following
information was found:
| 3^1 |
3^2 |
3^3 |
3^4 |
3^5 |
3^6 |
| 3 |
9 |
27 |
81 |
243 |
729 |
| 3^-1 |
3^-2 |
3^-3 |
3^-3 |
3^-5 |
3^-6 |
| 1/3 |
1/9 |
1/27 |
1/81 |
1/243 |
1/729 |
The number line is written out in the two tables above.
The first table has numbers that
approach, but never touch, zero from the positive direction (right to left on
the number
line). The second table has numbers that approach, but never touch zero from the
negative direction (left to right on the number line). As the exponent
approaches zero,
the answer also approaches zero. Our intuitive side says that when the exponent
is zero,
so is the answer.
The more formal proof uses the laws of exponents . The
fol lowing is the proof :
for all n, x, and y. So for example,
Now suppose we have the fraction:
This fraction equals 1, because the numerator and the
denominator are the same. If we
apply the law of exponents, we get:
So 3^0 = 1. ( math teacher website, 2008)
We can plug in any in number in the place of three, and that number raised to
the zero
power will still be 1.
Other Questions that Occurred
• How could people make such huge, interesting leaps as to come up with
exponents?
• Are there more people who discovered exponents that were not mentioned
because they might not have been rich or ambitious enough to publish their
findings?
• What fun activities, besides the usual graphing , can I bring to my students so
that
they can relate to exponents? How can I make this idea relevant in my students’
lives?
Connections:
• Exponents is shorthand that everyone can use.
• We use exponents when we speak of area and volume (3in^2 could be used for
area and 3ft^3 could be used for volume)
• We can talk about extremely large or small numbers by using exponents.
Why I picked these questions:
I have always wondered why an exponent of zero would always produce an answer of
zero. When students have asked, I always said that I would look into it and then
never
had the time. This gave me the opportunity to research something that I was and
am
interested in so that I can have that background knowledge not only for myself,
but for
inquisitive students as well.
