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There Are Infinitely Many Prime Numbers
Euclid’s famous proof of this is shown here.

It’s a proof by contradiction: assume there are only a finite number of prime numbers, and we have the list of them. Euclid shows that we can always use that list to find another prime that is not in the original list. This contradicts the original assumption, so that the alternative must be true: there are an infinitude of prime numbers.

Mathematician G. H. Hardy said “This proof is as fresh and significant as when it was discovered–two thousand years have not written a wrinkle on it”

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Mental Math – Squaring a Number Near 100

1. Subtract 100 from the number. Call this d.
2. Add d to the original number.
3. Append d ²

Example: 109²
1. d = 109 - 100 = 9
2. 109 + 9 = 118
3. d ² = 81, so answer = 11881

Example: 92²
1. 92 - 100 = -8
2. 92 + (-8) = 84
3. d ² = 64, so answer = 8464

Obviously, this works only for numbers in the range 91 to 109.
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The Harmonic Series is Infinite
Most of the infinite series that we meet have a finite sum. For example, in high school mathematics, we learn that 1 + 1/2 + 1/4 + 1/8 + 1/16 + … has a sum of 2 (geometric series).

This seems possible – though there are an infinite number of terms, the terms become infinitely small.

However, a surprise awaits with the series 1 + 1/2 + 1/3 + 1/4 + … Called the harmonic series, this sum can be proven to be infinite, even though the terms are becoming infinitely small. Roughly speaking, the terms do not become small fast enough.

The fact that the sum is infinite was proven in the year 1350 by Nicole Oresme. This proof is included in a list of the 100 greatest mathematics theorems, and it uses only basic arithmetic.

In his proof, shown in the graphic, he writes another infinite series which has two properties. First, its sum is larger than that of the harmonic series. Second, the new series has an infinite sum because it is equivalent to an infinite numbers of terms, each with a value of 1/2. Therefore the harmonic series is also infinite.

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Today, 6-28, is a Perfect Day
June 28, or 6-28, is a “perfect day” because both 6 and 28 are so-called perfect numbers.

A perfect number is one whose divisors add up to itself. The divisors of 6 are 1, 2, and 3, and 1+2+3=6. Likewise, 28=1+2+4+7+14. After these two, the next three perfect numbers are 496, 8128, and 33550336.

2,300 years ago, Euclid discovered a way to generate perfect numbers. First, we find a number n such that 2ⁿ – 1 is a prime number. Then, that prime multiplied by 2ⁿ⁻¹ is a perfect number. For example, n = 3 gives 2³ – 1 = 7, which is prime. So, 7 x 2² = 28 is a perfect number. Such perfect numbers are always even.

In 1755, Leonard Euler proved that there can be no even perfect numbers except for those that Euclid’s formula generates. However, he could not determine whether there any perfect numbers that are odd. 261 years later, this question is still unresolved.

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Unresolved Problems in Mathematics – the Twin Prime Conjecture
About 2,300 years ago, Euclid proved that there must be an infinite number of prime numbers. He also noted that many prime numbers differ by only 2, for example, 3 and 5, 11 and 13, 71 and 73, etc. These he called twin primes, but he offered no proof that there are an infinite number of them.

The assertion that there are infinitely many twin primes is called the twin prime conjecture. No proof for it has been found, making it perhaps the oldest unresolved problem in mathematics. Numerical evidence suggests that it is true. As prime numbers become larger, the fraction of them that are part of a twin becomes smaller, but the rate of decrease seems to level out.

In the last three years, some progress has been made. There is now a proof that there are an infinite number of prime pairs that differ by no more than 240.

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Ramanujan’s Partition Formula
The movie about mathematician Srinivasa Ramanujan, The Man Who Knew Infinity, has just been released, and this seems like a good time to mention one of the areas in which Ramanujan made breakthroughs, namely, the theory of partitions.

For those who don’t know, a partition of an integer n is a set of positive integers whose sum equals n. For example, one partition of 4 is simply 2 + 1 + 1. The partition function, p(n), is defined to be the number of such partitions of integer n. For n = 4, there are 4 partitions:
3 + 1, 2 + 2, 2 + 1 + 1, and 1 + 1 +1 + 1.
So p(4) = 4.

Obviously, the value of the function gets larger as n increases. One can list the possibilities, and laboriously verify that p(10) = 42, and p(20) = 627. However, for 150 years, mathematicians were unable to find an explicit formula for the function.

Then, in 1918, Ramanujan produced the formula shown in the graphic. Except for a few mathematicians, the formula is more amazing than enlightening. One can only wonder what square roots, pi, e, etc. could have to do with p(n).

Had Ramanujan not died at age 32, there would probably be many more such results.

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Viviani’s Theorem
The ancient Greeks discovered a lot of geometry, but they also missed a few things. One of these is Viviani’s Theorem, discovered in 1659.

Choose any point inside an equilateral triangle. The sum of the distances from the point to the three sides is equal to the triangle’s altitude.

In the animation, the dotted line is the altitude. The three distances always sum to the length of the altitude.

The second graphic shows the surprisingly simple proof of the theorem. The equilateral triangle is made up of three smaller triangles, which meet at the interior point. The total area of the three triangles is equal to the area of the equilateral triangle. When this equality is written as an equation, and common values cancelled, we have the theorem.

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Unsolved Problems in Mathematics – the Goldbach Conjecture
Every even number greater than 2 is equal to the sum of two prime numbers.

Examples:
8= 3 + 5
24 = 5 + 19, 7 + 17, 11 + 13
38 = 7 + 31
64 = 3+61, 5+59, 11+53, 17+47, 23+41

Christian Goldbach stated the conjecture in a 1742 letter to the great mathematician Leonard Euler.

Euler said that it seemed to be correct, but that he was unable to prove it. And so it has remained for the past 274 years, as many mathematicians have tried, and failed, to establish that it is true in general.

A related problem is usually called Goldbach’s weak conjecture, which was proven in 2013:

Every odd number greater than 5 is the sum of three prime numbers.

This is the first of series of “unsolved problems” posts.

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Pi
Today, March 14, or 3-14, is pi day. I offer this animation to add to the day’s festivities.

Pi, of course, has an infinite number of decimal places. Here are some notes on the history of efforts to compute its value with ever greater precision.

Archimedes was the first to actually calculate a value for pi. He used a circle of diameter 1, which, by definition, has a circumference of pi. To approximate the circumference, Archimedes imagined a 96-sided polygon inscribed inside this circle, and he computed its perimeter to be 3 1/7. This perimeter is slightly less than the circle’s circumference, so it is a lower bound for the value of pi. He did the same for a 96-gon on the outside of the circle, and obtained a perimeter of 3 11/70, the upper bound. Archimedes method was ingenious, but very laborious.

In 1615, Ludolph van Ceulen used a polygon of 32 212 254 720 sides to compute pi to 32 decimal places. He spent seven years doing the calculations.

In 1665, Isaac Newton discovered a non-geometric method to calculate pi – an infinite series. This was vastly more efficient, and he computed pi to 16 decimal places in only a few hours of work.

By 1949, pi had been calculated to an absurd 1,120 decimal places. After that, computers took over, and the current record is 13,300,000,000,000 decimal places.

Animation source: Wolfram Demonstrations

#mathematics #piday



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This image, made up of 9,000 ellipses, is by mathematician Hamid Naderi Yeganeh.

mathematics.culturalspot.org
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