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In a previous post I discussed Rational and Irrational Numbers.
Today I want to discuss numbers that are Prime.
In today’s slang, Prime means something that is cool, sweet, or
awesome.[i]
When discussing Prime Numbers many mathematicians who would agree
this definition of Prime is fitting for describing Prime Numbers.
When you look for an official definition of a Prime Number you
can find many. Most will define a Prime Number as a number divisible by 1 or
itself. More complete definitions will include the information that a Prime
Number must be a whole number greater than 1.[ii]
The definition I prefer is, “a prime number is a whole number
greater than 1, whose only two whole-number factors are 1 and itself.”[iii]
This definition includes all the key information necessary to understand what a
Prime Number is. It also makes clear the fact that the number 1 is not a Prime
Number. This is true because Prime Numbers have exactly two whole number
factors and 1 has only itself as a factor.
Figuring out what numbers are Prime has evolved into something
of an international sport. There are even organizations offering prizes for the
discovery of the newest and largest prime numbers. Because numbers are
infinite, the number of Prime Numbers is also infinite. As computing power
increases so will mathematicians' ability to discover ever larger Prime
Numbers.
Most students can name the first half-dozen or so Prime Numbers
off the top of their heads. For example: 2, 3, 5, 7, 11, and 13 are the first
six Prime Numbers. After 13 they become harder to remember. Two thousand three
hundred and some odd years ago a Greek mathematician named Eratosthenes came up
with a way to quickly identify all the Prime Numbers below 100. His method
became known as the Eratosthenes’ Sieve. My students always enjoy this lesson
because they enjoy crossing things out and they enjoy doing things that seem
almost magical.
If you’d like to know more about Eratosthenes’ Sieve you can
visit The Prime Glossary’s Sieve of
Eratosthenes page. The MATHCOUNTS
NOTES blog also has some excellent information on Prime Numbers and some of
the vocabulary associated with them.
Even textbook publishers can get confused about Prime Numbers.
The textbook I used in class several years ago gave 9 as an answer to a
question about what Prime Numbers divide evenly into 36. My students were
tickled to learn that even the “experts” get things wrong sometimes.
One of my favorite bonus questions to include on a test after I
teach about Prime Numbers is to challenge the students to make a list of all
the even Prime Numbers. Those who have paid attention in class will know the
only even Prime Number is 2, because all other even numbers have 1, themselves,
and 2 as factors. Any number with three or more factors cannot be a Prime
Number.
There is much more to the set of Prime Numbers to be learned
and, as I mentioned earlier, the search for the next largest Prime Number is
ongoing. The largest Prime Number, as of January 2016, is acknowledged to be 274,207,281
– 1.[iv]
To calculate this Prime Number as a standard number, multiply 2 times itself
74,207,281 times and subtract 1 from the final product. The resulting number
would be 22,338,618 digits long.[v]
Such Prime Numbers are known as Mersennes Primes and only forty-nine have yet
been discovered and proven though the search for such Prime Numbers dates back
to Euclid around 350 BC.[vi]
When it comes to my students, while I will tell them about the
ongoing search for the next largest Prime Number, we will stick to using Eratosthenes’
Sieve to find the Prime Numbers between 1 and 100.
As always, I remain,
The Exhausted Educator
[iii] Rouse, Margaret.
"What Is Prime Number? - Definition from WhatIs.com." WhatIs.com.
TechTarget, June 2015. Web. 03 Aug. 2016.
[iv] "Mersenne
Prime Discovery - 2^74207281-1 Is Prime!" Mersenne Prime Discovery -
2^74207281-1 Is Prime! Mersenne Research, Inc., Jan. 2016. Web. 03 Aug.
2016.
[v]
ibid
[vi]
ibid
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