- 3884 - NUMBERS - that are special formulas? - The world's most famous irrational number, pi, whose first 10 digits are 3.141592653. As 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.
---------------
3884 - NUMBERS
- that are special formulas?
-
---------------------------- Tau
---------------------------- Natural log base
---------------------------- Imaginary number i
---------------------- i to the power of i
------------------------ Belphegor's prime number
------------------------ 2^{aleph_0}
------------------------ Apéry's constant
------------------------- The number 1
------------------------- Euler's identity
------------------------- The number 0
-------------------------- The square root of 2
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- Two times
pi, or the number "tau," which is roughly 6.28. 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.
-
- The base
of natural logarithms — written as "e" for its namesake, the
18th-century Swiss mathematician Leonhard Euler. 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.
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- 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. 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.
-
- The number
“i” is a special 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.
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- If we break
that rule, we get to invent the “imaginary numbers”, and so the complex
numbers, which are both beautiful and useful.
Complex numbers can be expressed as the sum of both real and imaginary
parts.
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- The
imaginary number i is an exceptionally weird number because -1 has two square
roots: i and -i. “i” to the power of
“i”. At a glance, this looks like the
most imaginary number possible, an imaginary number raised to an imaginary
power. Leonhard Euler wrote in a 1746
letter, it is a “real number!”
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- Finding the
value of i to the i power involves rearranging Euler's identity, a formula
relating the irrational number e, the imaginary number i, and the sine and
cosine of a given angle. When you solve the formula for a 90-degree angle
(which can be expressed as pi over 2), you can simplify the equation to show
that i to the power of i equals e raised to the power of negative pi over 2.
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- The result
equals roughly 0.207. It is a very real
number in the case of a 90-degree angle.
As Euler pointed out, i to the i power does not have a single
value. It takes on "infinitely
many" values depending on the angle you're solving for.
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- Belphegor's
prime number is a palindromic prime number with a 666 hiding between 13 zeros and
a 1 on each 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.
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- Belphegor (or
Beelphegor) is one of the seven demon princes of hell in the Bible. The number apparently even has its own
devilish symbol, which looks like an upside-down symbol for pi.
-
- Harvard
mathematician W. Hugh Woodin has devoted many years of research to infinite
numbers. It's no surprise, then, that his favorite number is an infinite one: 2^{aleph_0},
or 2 raised to the power of aleph-naught, also called aleph-null.
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- Aleph
numbers are used to describe the sizes of infinite sets, where a set is any
collection of distinct objects in mathematics.
For example, the numbers 2, 4 and 6 can form a set of size 3. In other words, there's always something
bigger: Infinite cardinal numbers are infinite, so there is no such thing as
the "largest cardinal number."
-
- 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 with 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,
would allow mathematicians to better predict how the prime numbers are
distributed.
-
- It turns out
that Apéry's constant shows up in fascinating places in physics, including in
equations governing the electron's magnetism and orientation to its angular
momentum.
-
- 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. 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, which is actually an equation.
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.
-
- If we're
already talking about how awesome 1 is, then why not throw in the even weirder
and cooler number 0? For most of written human history, the concept of zero
wasn't all that important.
-
- The ancient
Greeks began to develop the idea of using zero as an empty place indicator to
distinguish numbers of different magnitudes, but it wasn't until roughly the
seventh century that Indian mathematicians, like Brahmagupta, began describing
the modern idea of zero.
-
-
Brahmagupta wrote that any number multiplied by zero is zero, but he
struggled with division, saying that a number, n, divided by zero just comes
out as n/0, rather than the modern answer, which is that the result is
undefined. The Maya had also independently derived the concept of zero by A.D.
665.
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- Perhaps the
most dangerous number ever conceived, the square root of 2 supposedly led to the
first mathematical murder in history. The Greek mathematician Hippasus of
Metapontum is credited with discovering it in the fifth century B.C. While
working on a separate problem, Hippasus is said to have stumbled on the fact
that an isosceles right triangle whose two base sides are 1 unit in length will
have a hypotenuse that is √2, which is an irrational number.
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- According
to legend, Hippasus' contemporaries, members of the quasi-religious order known
as the Pythagoreans, threw him into the sea after hearing about his great
discovery. That's because the Pythagoreans believed that "all is
number" and the universe only contained whole numbers and their ratios.
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- Irrational
numbers like √2 (and pi), which can't be expressed as a ratio of whole numbers
and go on forever after the decimal place, were seen as an abomination.
-
- These days,
we're a little calmer about √2, often calling it Pythagoras' constant. It
starts off as 1.4142135623 … and goes on forever.
-
February 24,
2023 NUMBERS - that
are special formulas? 3884
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