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Mathematical Constants

1. Pi

This is the most well known mathematical constant, and is the ratio of the circumference of a circle to its diameter - the fact that this is the same for all circles is amazing in itself.

The valuepi has been estimated since biblical times, Pi through the ages, and a (wrong) value was almost written into a law.

You will know that pi can be approximated by22/7 which only gives the value correct to 2 decimal places. 355/113 is correct to 6 decimal places, and103993/33102to 9 decimal places. pi has been calculated to over 50 billion digits. See Table of current records for the computation of constants for the current record forpi and other constants.

pi is not only irrational (it cannot be written exactly as the ratio of two integers) it is also transcendental (it is not the solution of any polynomial equation with rational coefficients). You will find proofs in Transcendence of Pi and in TEX format. Not many transcendental numbers are known (e and Liouville's number are another two examples) but in fact in 1874 Cantor showed that almost all real numbers are transcendental. You can see some of the proofs if you follow the links at Foundations.

There are lots of formulae for pi. A couple appear in formulae and here are two more:

This formula was discovered by François Viète:

Viete's formula for pi

and Isaac Newton used 22 terms of this series to find pi correct to 16 places:

Newton's formula for pi

pi appears everywhere and you will find it popping up in many of the pages of this site.

 It is the subject of a film called, naturally, Pi. See Pi The Movie and the excellent Internet Movie Database.

A good page to start your researches intopi is The Pi Pages though you may want to turn down the sound since it recites the digits of pi in French - don't ask why!

How Pi is calculated shows how to calculate pi.

Useless facts about Pi is what it says it is.

Pi Facts gives 71 facts aboutpi.

The Ridiculously Enhanced Pi Page is worth a visit

There are lots of really daft sites devoted to pi like Pi Day Songs.

You'll find formulae involvingpiat Expansions for PI.

pi attracts a lot of attention and that is reflected in the huge number of sites on the internet. You can spend lots of happy days researching these sites.

2. e

It is harder to give a simple definition for e like that forpi; e is usually introduced at A level. One way to define it is to say that it is the number that makes the shaded area in the graph of y = 1/xequal to 1.

graph defining e

The value of e is 2.718 281 828 459 045 ... and it has the unusual attribute that the numbers 1828 repeat early on in the decimal expansion. e is another example of a transcendental number (see Liouville).

Its series expansion

Series for e

is well known, and converges quickly, giving good estimates for e.

That would be the end of the matter if it wasn't for the fact that e likepi keeps popping up in all sorts of unexpected places. You will find it occurs in lots of places on this site, most famously along with pi.

One of the first properties that students meet is that the rate of change of ex is ex, so it's exceptionally easy to differentiate or integrate, and its use in calculus is important.

In addition, it occurs in formulae that deal with anything from population increase to radioactive decay. Exponential Growth: The Andromeda Strain, Growth and Decay (do try Problem 1 on that page).

It occurs in interest calculations: you should know that getting interest at at 6% twice a year yields more than 12% per year. Interest added at 1% per month is even better. If you keep increasing the frequency of adding interest to daily, hourly, minutely (is there such a word?) etc you will get a higher return but there's a limit and that limit is e. This is because

Limit formula for e

See Compound Interest and Math Buffs Find an Easier "e".

It also pops up in probability. For example, if you have a number of letters and the same number of envelopes, then the probability that every letter is placed in the wrong envelope approaches1/e as the number of letters and envelopes increases. This is the same problem as turning up pairs of cards from two packs, with the probability that there is no match among the 52 pairs being approximately1/e. Paychecks and Envelopes

Finally, can you find the value of e to power (pi times root 163)? This number is interesting because it appears to be a large integer but it isn't. So how many decimal places are required before you can see that it isn't an integer?

Useful pages on e are:

The Euler's Number Home Page

The Natural Logarithmic Base

Some History Surrounding e

Why is e so important?

3. i

i stands for imaginary number because it seemed like that when first used. Yet it is no more imaginary than negative numbers (how often have you seen -2 pieces of cake?) or irrational numbers likeroot 2, which Pythagoras had trouble with. Indeed it was the use of i that revolutionised the use of electricity enabling the design of dynamos and electric motors Applications of Complex Numbers , Real Life Applications of Imaginary Numbers. Mathematicians used i long before they understood it, because they found it was so useful.

i arose naturally when trying to solve equations like x2 + 1 = 0. Using i and complex numbers like 2 + 3i means that one can say that all quadratic equations have two solutions (though they may be equal). Indeed the Fundamental Theorem of Algebra (you'll find proofs at Intuition and Rigor) says that every polynomial of degree n has n roots (root is another word for solution). The set of complex numbers Complex numbers symbolis then the top of the chain leading from natural numbers to integers to rationals to reals and finally to the complex numbers (though see Number Systems). Complex numbers symbolis then called algebraically closed in mathematical jargon Algebraic structures.

In the sixteenth century, Cardan (also known as Cardano), who was one of the first to give a formula for solving a cubic, was able to solve the problem of dividing 10 into two parts such that the product is 40, by giving the answer as 5 + root(-15), 5 - root(-15) but he wasn't happy with these 'truly sophisticated' numbers.

i is fun to play around with, so for example

powers of i

i reveals an unexpected link between e and the trigonometric functions sin and cos thus unifying two areas of mathematics. This leads to simple proofs of trig identities and results like De Moivre's theorem Expansion of sin and cos using De Moivre's theorem

De Moivre's formula

which appears more naturally as

(e^ix)^n=e^inx

using the unexpected link.

Unlike many of the constants on this page there is no decimal equivalent nor is it possible to compare the size of i with these constants. Think about why this is so.

One does have to be careful when using i because it doesn't behave quite like a real number 1=2: A Proof using Complex Numbers.

It is a shame that complex numbers have dropped out of some A level syllabuses because they open the door to a whole new world Basic Definitions, Introduction to complex numbers .

4. Euler's Constant gamma

It isn't difficult to show Testing For Convergence that the series Series of reciprocal powersconverges if s > 1 and diverges if s < 1. The sums of this series when s = 2, 4, 6 and 8 are given in Euler Zeta Function.

But what happens if s = 1? The answer is that it diverges but very slowly. In fact as N gets larger the series Series for gamma, called the harmonic series, gets closer and closer to log n plus gammawheregammais known as Euler's or Mascheroni's constant and is approximately 0.577 215 664 901 ... It is believed that gammais irrational but it hasn't been proved.

So how slowly does the harmonic series diverge? If N = 108 then log n plus gammais approximately 21.3. Thus the sum of one hundred million terms is only just over twenty. That's slow divergence for you!

Euler-Mascheroni Constant has lots more on gamma.

5. The Golden Ratio Phi

The golden ratiophihas the value(Root 5 + 1)/2which is 1.618 033 988 ... It is obviously irrational (why?) and algebraic (the opposite of transcendental) being one of the two solutions of x2 - x - 1 = 0. The other solution is 1/phiwhich is 0.618 033 988 ... (confusingly, this value is sometimes called the Golden Ratio). Notice that 1/phi = phi - 1.

phiis another one of those numbers that pops up in all sorts of surprising places. If you draw a rectangle with sides whose lengths are in the ratio phi: 1, then this rectangle is said to be aesthetically pleasing to the eye and many Renaissance artists used it in their paintings The Golden Mean, Golden Figures.

If you take a 'golden rectangle' Some Golden Geometry then it can be divided into a square plus a similar rectangle The Golden Spiral

Golden rectangle and spiral

The original rectangle gives rise to a new one in which the ratio of the sides is now 1 : (phi - 1)which is the same as phi: 1 (why?). Now keep dividing in this way and you can join up the vertices to get a logarithmic spiral, also known as an equiangular spiral. The Golden Bowls & the Logarithmic Spiral, Logarithmic Spirals (beautiful pictures of shells), Equiangular Spiral.

This spiral occurs frequently in nature, for example in the arrangement of sunflower heads and spiral shells Fibonacci Numbers and Nature.

phican be written as the continued fractionContunued fraction for phithough this converges very slowly. An Introduction to Continued Fractions.

It is also intimately connected to the Fibonnaci numbers Fibonacci Numbers and the Golden Section 1, 1, 2, 3, 5, 8, 13, 21, 34 where each term is the sum of the previous two. The continued fraction forphigives convergentscontinued fraction convergentswhose numbers are the Fibonacci numbers Continued fractions and the Fibonacci sequence. Furthermore, the nth Fibonacci number equals the nearest integer to 1/root5 times phi^n, and it follows that the ratio of successive Fibonnaci numbers tends tophi.

There are lots of sites aboutphiand they include:

Phi's Fascinating Figures makesphivery interesting.

Phi: That Golden Number includes lots of geometric pictures with Golden Triangles, Golden Ellipse, Whirling Triangles, Whirling Rectangles amongst them.

The Golden Mean includes lots of references.

 

6. Countable cardinal Aleph 0

Mathematicians looked to the Greek alphabet for many of their symbols, withpi being the most common. Then they ran out of Greek letters and turned to the Hebrew alphabet for aleph aleph, to Gothic letters like Gothic letter, to double-lined Latin letters (called blackboard bold) like Real numbers symbol for the set of real numbers, to made up letters likefor all symboland, occasionally, to the Cyrillic alphabet.

So what isaleph zero? It is to do with infinite sets. Such sets can be difficult to deal with, so that it is awkward to compare sizes of infinite sets. The key is to use one-one correspondence. If you look in a classroom you can easily see whether there are more seats than students without counting. What you are doing is pairing off each filled seat with the student sitting on them then seeing if there are any seats (or students) left over. If there's no seat or student left over then you have established a one-one correspondence between the set of chairs and the set of students.

Cantor applied this principle to infinite sets and gave the setset of integers symbolof all integers a 'size' which is called cardinality, ofaleph zero. Any set which could be put into one-one correspondence withset of integers symbolalso has cardinalityaleph zeroand is called countable. This leads to interesting results. For example, each even number 2n can be paired with the integer n so the set of even numbers is also countable. Cantor showed that the set of rational numbersset of rationals symbol is also countable but that the set of real numbersReal numbers symbolis not. The question whether there is a cardinal number betweenaleph zeroand that ofReal numbers symbolis called the Continuum Hypothesis and it has been answered in a most surprising way.

The Continuum Hypothesis

Mudd Math Fun Facts: Continuum Hypothesis

A Crash Course in the Mathematics Of Infinite Sets

Infinite Reflections is all about infinity.

Hilbert's Hotel will make you think about infinity.

7. root 2

This is the number that's used to introduce students to the idea of an irrational number. A rational number is expressible as the ratio of two integers. All finite decimals (like 1.23456) and repeating decimals (like 0.3333...) are rational. Any real number that isn't rational is called irrational. (See Cantor.) Look at Pythagoras for links to proofs that root 2is irrational.

You can approximate toroot 2 by the sequence

sequence of approximations to root 2

and it can be shown that this is the best possible sequence of approximations. Can you work out how each term is obtained from the previous one?

8. Liouville's number Liouville's number

In 1844 Liouville showed that the number

Liouville's number

was transcendental, and this was the first such number known. It was a struggle to provepiand e were also transcendental (Hermite proved e was transcendental in 1873 and Lindemann provedpiwas transcendental in 1882) and in 1874 Cantor shocked the mathematical world by showing that almost all real numbers are transcendental, so if you pick a real number at random it is virtually certain to be transcendental. Common they may be, but very few actual examples are known. The 15 Most Famous Transcendental Numbers.

9. Googol and Googolplex

The mathematician Edward Kasner's 9-year-old nephew coined the names googol for the number that is 1 followed by a hundred zeroes, 10100 and googolplex for 1 followed by a googol of zeroes, 10googol.

These numbers are very large and impossible to visualise. A googol is larger than any known estimate of the number of atoms in the universe and thus it would be physically impossible to write out the even larger googolplex since there are aren't the atoms available to make up the zeroes.

The human mind finds it difficult to comprehend such vast numbers. To illustrate this, guess how long you think it would take to count from one to a million, which is a mere 106, counting at a rate of one number per second without a break. How long would it take to count to a billion (US version) 109? Check your guesses by using a calculator (think a for minute and should be clear how to do this).

Humour : What's a GooGol?

Large Numbers - Notes

The Googolplex Page is an amusing look at googolplex.

10. Graham's number

This is an outrageously large number which puts googol and googolplex in the shade. It is so large that a special notation has to be invented in order to describe it. It is actually an estimate for a number that occurs in combinatorics and the joke is that the number is believed to be 6, so Graham's number must rank as one of the worst estimates ever.

Graham's Number gives a good description of the number.

Large Numbers

Big Numbers.

 

You'll find more constants at Favorite Mathematical Constants, a few more at Some Important Constants and at RJN's More Digits of Irrational Numbers Page, which gives the decimal expansions of many of them to millions of digits.

 

 

Steve Mayer

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