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9 Numbers That Are Cooler Than Pi

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Pi is an irrational number, meaning it cannot be written as a simple fraction.

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It's March 14, and that means only one thing … it's Pi Day and time to celebrate the world's most famous irrational number, pi. The ratio of a circle's circumference to its diameter, pi is not just irrational, meaning it can't be written as a simple fraction; it is also transcendental, meaning it's not the root, or solution, to any polynomial equation, such as x+2X^2+3 = 0.

But no so fast … pi may be one of the most well-known numbers, but for people who are paid to think about numbers all day long, the circle constant can be a bit of a bore. In fact, countless numbers are potentially even cooler than pi. We asked several mathematicians what their favorite post-pi numbers are; here are some of their answers.

Tau

Two gooseberry pies.

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You know what's cooler than ONE pie? … TWO pies. In other words, two times pi, or the number "tau," which is roughly 6.28.

"Using tau makes every formula clearer and more logical than using pi," said John Baez, a mathematician at the University of California, Riverside. "Our focus on pi rather than 2pi is a historical accident."

Tau is what shows up in the most important formulas, he said.

While pi relates a circle's circumference to its diameter, tau relates a circle's circumference to its radius — and many mathematicians argue that this relationship is much more important. Tau also makes seemingly unrelated equations nicely symmetrical, such as the one for a circle's area and an equation describing kinetic and elastic energy.

But tau will not be forgotten on pi day! As per tradition, the Massachusetts Institute of Technology will send out decisions at 6:28 p.m. today. A few months from now, on June 28, tau will have its own day.

Natural log base

Natural log

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The base of natural logarithms — written as "e" for its namesake, the 18th-century Swiss mathematician Leonhard Euler — may not be as famous as pi, but it also has its own holiday. Yup, while 3.14 is celebrated on March 14, natural log base, the irrational number beginning with 2.718, is lionized on Feb. 7.

The base of natural logarithms is most often used in equations involving logarithms, exponential growth and complex numbers.

"[It] has the wonderful definition as being the one number for which the exponential function y = e^x has a slope equal to its value at every point," Keith Devlin, the director of the Stanford University Mathematics Outreach Project in the Graduate School of Education, told Live Science. In other words, if the value of a function is, say 7.5 at a certain point, then its slope, or derivative, at that point is also 7.5. And, "like pi, it comes up all the time in mathematics, physics and engineering."

Imaginary number i

Imaginary number i.

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Take the "p" out of "pi," and what do you get? That's right, the number i. No, that's not really how it works, but i is a pretty cool number. It's the square root of -1, which means it's a rule breaker, as you're not supposed to take the square root of a negative number.

"Yet, if we break that rule, we get to invent the imaginary numbers, and so the complex numbers, which are both beautiful and useful," Eugenia Cheng, a mathematician at the School of the Art Institute of Chicago, told Live Science in an email. (Complex numbers can be expressed as the sum of both real and imaginary parts.)

i is an exceptionally weird number, because -1 has two square roots: i and -i, Cheng said. "But we can't tell which one is which!" Mathematicians have to just pick one square root and call it i and the other -i.

"It's weird and wonderful," Cheng said.

i to the power of i

i to the power of i

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Believe it or not, there are ways to make i even weirder. For example, you can raise i to the power of i — in other words, take the square root of -1 raised to the square-root-of-negative-one power.

"At a glance, this looks like the most imaginary number possible — an imaginary number raised to an imaginary power," David Richeson, a professor of mathematics at Dickinson College in Pennsylvania and author of the forthcoming book "Tales of Impossibility: The 2,000-Year Quest to Solve the Mathematical Problems of Antiquity," (Princeton University Press), told Live Science. "But, in fact, as Leonhard Euler wrote in a 1746 letter, it is a real number!"

Finding the value of i to the i power involves rearranging Euler's formula relating the irrational number e, the imaginary number i, and the sine and cosine of a given angle. When solving the formula for a 90-degree angle (which can be expressed as pi over 2), the equation can be simplified to show that i to the power of i equals e raised to the power of negative pi over 2.

It sounds confusing (here's the full calculation, if you dare to read it), but the result equals roughly 0.207 — a very real number. At least, in the case of a 90-degree angle.

"As Euler pointed out, i to the i power does not have a single value," Richeson said, but rather takes on "infinitely many" values depending on the angle you're solving for. (Because of this, it's unlikely we'll ever see "i to the power of i day" celebrated as a calendar holiday.)

Belphegor's prime number

Depiction of the demon Belphégor, a demon of discovery and of ingenious inventions who often takes the form of a young woman, from J.A.S. Collin de Plancy.

(Image credit: Louis Le Breton/Dictionnaire Infernal)

Belphegor's prime number is a palindromic prime number with a 666 hiding between 13 zeros and a 1 on either side. The ominous number can be abbreviated as 1 0(13) 666 0(13) 1, where the (13) denotes the number of zeros between the 1 and 666.

Although he didn't "discover" the number, scientist and author Cliff Pickover made the sinister-feeling number famous when he named it after Belphegor (or Beelphegor), one of the seven demon princes of hell.

The number apparently even has its own devilish symbol, which looks like an upside-down symbol for pi. According to Pickover's website, the symbol is derived from a glyph in the mysterious Voynich manuscript, an early 15th-century compilation of illustrations and text that no one seems to understand.

2^{aleph_0}

infinity-symbol-galaxy-02

Harvard mathematician W. Hugh Woodin has devoted his years and years of research to infinite numbers, and so unsurprisingly, he chose as his favorite number an infinite one: 2^{aleph_0}, or 2 raised to the power of aleph-naught. Aleph numbers are used to describe the sizes of infinite sets, where a set is any collection of distinct objects in mathematics. (So, the numbers 2, 4 and 6 can form a set of size 3.)

As for why Woodin chose the number, he said, "Realizing that 2^{aleph_0} is not \aleph_0 (i.e. Cantor's theorem) is the realization that there are different sizes of infinite. So that makes the conception of 2^{\aleph_0} rather special."

In other words, there’s always something bigger: Infinite cardinal numbers are infinite, and so there is no such thing as the "largest cardinal number."

Apéry's constant

Atomic nucleus with orbiting electrons.

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"If naming a favorite, then the Apéry's constant (zeta(3)), because there is still some mystery associated with it," Harvard mathematician Oliver Knill told Live Science.

In 1979, French mathematician Roger Apéry proved that a value that would come to be known as Apéry's constant is an irrational number. (It begins 1.2020569 and continues infinitely.) The constant is also written as zeta(3), where "zeta(3)" is the Riemann zeta function when you plug in the number 3.

One of the biggest outstanding problems in math, the Riemann hypothesis, makes a prediction about when the Riemann zeta function equals zero, and if proven true, would allow mathematicians to better predict how the prime numbers are distributed.

Of the Riemann hypothesis, renowned 20th-century mathematician David Hilbert once said, "If I were to awaken after having slept for a thousand years, my first question would be, 'Has the Riemann hypothesis been proven?'"

So what's so cool about this constant? It turns out that Apéry's constant shows up in fascinating places in physics, including in equations governing the electron's magnetic strength and orientation to its angular momentum.

The number 1

Floating 3D number 1s.

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Ed Letzter, a mathematician at Temple University in Philadelphia (and, full disclosure, the father of Live Science staff writer Rafi Letzter), had a practical answer:

"I suppose this is a boring answer, but I'd have to choose 1 as my favorite, both as a number and in its different roles in so many different more abstract contexts," he told Live Science.

One is the only number by which all other numbers divide into integers. It's the only number divisible by exactly one positive integer (itself, 1). It's the only positive integer that's neither prime nor composite.

In both math and engineering, values are often represented as between 0 and 1. "One hundred percent" is just a fancy way of saying 1. It's whole and complete.

And of course, throughout the sciences, 1 is used to represent basic units. A single proton is said to have a charge of +1. In binary logic, 1 means yes. It's the atomic number of the lightest element, and it's the dimension of a straight line.

Euler's identity

leonhard euler

(Image credit: Jakob Emanuel Handmann/Wikimedia Commons)

Euler's identity, which is actually an equation, is a real mathematical jewel, at least as described by the late physicist Richard Feynman. It has also been compared to a Shakespearean sonnet.

In a nutshell, Euler's Identity ties together a number of mathematical constants: pi, natural log e and the imaginary unit i.

"[It] connects these three constants with the additive identity 0 and the multiplicative identity of elementary arithmetic: e^{i*Pi} + 1 = 0," Devlin said.

You can read more about Euler's Identity here.

Originally published on Live Science.