These are constants which one is likely to encounter during pre-college education in many countries.
Pythagoras' constant of a
right-angled triangle with legs of length 1. The
square root of 2, often known as
root 2 or '''Pythagoras' constant
, and written as , is the unique positive real number that, when multiplied by itself, gives the number 2. It is more precisely called the principal square root of 2''', to distinguish it from the
negative number with the same property. Geometrically the
square root of 2 is the length of a diagonal across a
square with sides of one unit of length; this follows from the
Pythagorean theorem. It is an
irrational number, possibly the first number to be known as such, and an
algebraic number. Its numerical value
truncated to 50
decimal places is: : . Alternatively, the quick approximation 99/70 (≈ 1.41429) for the square root of two was frequently used before the common use of
electronic calculators and
computers. Despite having a
denominator of only 70, it differs from the correct value by less than 1/10,000 (approx. 7.2 × 10−5). Its simple
continued fraction is periodic and given by: \sqrt2 = 1 + \frac{1}{2+\frac{1}{2+\frac{1}{2+\frac1\ddots}}}
Archimedes' constant is . The constant pi| (pi) has a natural
definition in
Euclidean geometry as the ratio between the
circumference and
diameter of a circle. It may be found in many other places in mathematics: for example, the
Gaussian integral, the complex
roots of unity, and
Cauchy distributions in
probability. However, its ubiquity is not limited to
pure mathematics. It appears in many formulas in physics, and several
physical constants are most naturally defined with or its reciprocal factored out. For example, the ground state
wave function of the
hydrogen atom is : \psi(\mathbf{r}) = \frac{1}{\sqrt{\pi{a_0}^3}} e^{-r/a_0}, where a_0 is the
Bohr radius. is an
irrational number,
transcendental number and an
algebraic period. The numeric value of is approximately: : . Unusually good approximations are given by the fractions
22/7 and
355/113.
Memorizing as well as
computing increasingly more digits of is a world record pursuit.
Euler's number Euler's number , also known as the
exponential growth constant, appears in many areas of mathematics, and one possible definition of it is the value of the following expression: :e = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n The constant is intrinsically related to the
exponential function x \mapsto e^x. The
Swiss mathematician
Jacob Bernoulli discovered that arises in
compound interest: If an account starts at $1, and yields interest at annual rate , then as the number of compounding periods per year tends to infinity (a situation known as
continuous compounding), the amount of money at the end of the year will approach dollars. The constant also has applications to
probability theory, where it arises in a way not obviously related to exponential growth. As an example, suppose that a slot machine with a one in probability of winning is played times, then for large (e.g., one million), the
probability that nothing will be won will tend to as tends to infinity. Another application of , discovered in part by Jacob Bernoulli along with
French mathematician
Pierre Raymond de Montmort, is in the problem of
derangements, also known as the
hat check problem. Here, guests are invited to a party, and at the door each guest checks his hat with the butler, who then places them into labelled boxes. The butler does not know the name of the guests, and hence must put them into boxes selected at random. The problem of de Montmort is: what is the probability that
none of the hats gets put into the right box. The answer is :p_n = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+(-1)^n\frac{1}{n!} which, as tends to infinity, approaches . is an irrational number and a transcendental number. The numeric value of is approximately: : .
The imaginary unit . Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis. The
imaginary unit or
unit imaginary number, denoted as , is a
mathematical concept which extends the
real number system to the
complex number system. The imaginary unit's core property is that . The term "
imaginary" was coined because there is no (
real) number having a negative
square. There are in fact two complex square roots of −1, namely and , just as there are two complex square roots of every other real number (except
zero, which has one double square root). In contexts where the symbol is ambiguous or problematic, or the Greek
iota () is sometimes used. This is in particular the case in
electrical engineering and
control systems engineering, where the imaginary unit is often denoted by , because is commonly used to denote
electric current.
The golden ratio F_n=\frac{\varphi^n-(1-\varphi)^n}{\sqrt 5} An explicit formula for the th
Fibonacci number involving the
golden ratio The number , also called the
golden ratio, turns up frequently in
geometry, particularly in figures with pentagonal
symmetry. Indeed, the length of a regular
pentagon's
diagonal is times its side. The vertices of a regular
icosahedron are those of three mutually
orthogonal golden rectangles. Also, it is related to the
Fibonacci sequence, related to growth by
recursion.
Kepler proved that it is the limit of the ratio of consecutive Fibonacci numbers. The golden ratio has the slowest converging
continued fraction of any irrational number. It is, for that reason, one of the
worst cases of
Lagrange's approximation theorem and it is an extremal case of the
Hurwitz inequality for
diophantine approximations to irrational numbers. This may be why angles close to the golden ratio often show up in
phyllotaxis (the growth of plants). It is approximately equal to: : . or, more precisely \frac{1+\sqrt{5}}{2}. == Constants in advanced mathematics ==