<![CDATA[Math Is Fun Forum / Euler Avenue]]> 2023-08-29T11:07:14Z FluxBB https://www.mathisfunforum.com/index.php <![CDATA[Euler's polyhedron formula]]> The relation in the number of vertices, edges and faces of a polyhedron gives Euler's Formula. By using Euler's Formula, V+F=E+2 can find the required missing face or edge or vertices.Euler's law states that  'For any real number x, e^ix = cos x + i sin x. This complex exponential function is sometimes denoted cis x ("cosine plus i sine"). The formula is still valid if x is a complex number. Abstract: “V − E + F = 2”, the famous Euler's polyhedral formula, has a natural generalization to convex polytopes in every finite dimension, also known as the Euler– Poincaré Formula. We provide another short inductive combinatorial proof of the general formula.For solid shapes, especially polyhedra, the sum of the faces and vertices will be 2 more than their edges. Faces + vertices = edges + 2. Another way of writing this is Faces + vertices - edges = 2. This is known as Euler's formula.

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https://www.mathisfunforum.com/profile.php?id=245364 2023-08-29T11:07:14Z https://www.mathisfunforum.com/viewtopic.php?id=26850&action=new
<![CDATA[Some special numbers]]> 56) Square Root of 5

The square root of 5 is the positive real number that, when multiplied by itself, gives the prime number 5. It is more precisely called the principal square root of 5, to distinguish it from the negative number with the same property. This number appears in the fractional expression for the golden ratio. It can be denoted in surd form as:

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.23606797749978969640917366873127623544061835961152572427089...
which can be rounded down to 2.236 to within 99.99% accuracy. The approximation
161/72  (≈ 2.23611) for the square root of five can be used. Despite having a denominator of only 72, it differs from the correct value by less than
1/10,000
approx.

As of January 2022, its numerical value in decimal has been computed to at least 2,250,000,000,000 digits.

The successive partial evaluations of the continued fraction, which are called its convergents, approach

:

Their numerators are 2, 9, 38, 161, … ,  and their denominators are 1, 4, 17, 72, … .

Each of these is a best rational approximation of

; in other words, it is closer to
than any rational with a smaller denominator.

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https://www.mathisfunforum.com/profile.php?id=682 2023-01-03T13:05:10Z https://www.mathisfunforum.com/viewtopic.php?id=25786&action=new
<![CDATA[What do you think is the worst way to learn mathematics?]]> I don't think there is worse way.

The best -

The three Is - Intuition, Intelligence, and Improvement - I think that is sufficient in Mathematics.

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https://www.mathisfunforum.com/profile.php?id=682 2022-10-25T13:04:23Z https://www.mathisfunforum.com/viewtopic.php?id=21960&action=new
<![CDATA[Two sequences and the exponential function]]> hi theshire

I'm seeing a lot of LaTex errors in this.  I hope you don't mind but I've had a go at editing some but I'm worried I might have changed your proof in the process.  Please have a look and see if it's ok. Also ask if you are unsure of the Latex that works for MIF. Not all feaures of the coding do, I'm sorry. What did you want list to do?  Some frac errors towards the end.

Bob

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https://www.mathisfunforum.com/profile.php?id=67694 2022-08-29T17:16:35Z https://www.mathisfunforum.com/viewtopic.php?id=27915&action=new
<![CDATA[Tetration]]> Tetration

In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though

and the left-exponent
are common.

Under the definition as repeated exponentiation,

means
, where n copies of a are iterated via exponentiation, right-to-left, I.e. the application of exponentiation n-1 times. n is called the "height" of the function, while a is called the "base," analogous to exponentiation. It would be read as "the nth tetration of a".

It is the next hyperoperation after exponentiation, but before pentation. The word was coined by Reuben Louis Goodstein from tetra- (four) and iteration.

Tetration is also defined recursively as

allowing for attempts to extend tetration to non-natural numbers such as real and complex numbers.

The two inverses of tetration are called super-root and super-logarithm, analogous to the nth root and the logarithmic functions. None of the three functions are elementary.

Tetration is used for the notation of very large numbers.

Introduction

The first four hyperoperations are shown here, with tetration being considered the fourth in the series. The unary operation succession, defined as

, is considered to be the zeroth operation.

n copies of 1 added to a.

Multiplication

n copies of a combined by addition.

Exponentiation

n copies of a combined by multiplication.

Tetration

n copies of a combined by exponentiation, right-to-left.

Succession, (a′ = a + 1), is the most basic operation; while addition (a + n) is a primary operation, for addition of natural numbers it can be thought of as a chained succession of n successors of a; multiplication (a × n) is also a primary operation, though for natural numbers it can analogously be thought of as a chained addition involving n numbers of a. Exponentiation can be thought of as a chained multiplication involving n numbers of a and tetration

as a chained power involving n numbers a. Each of the operations above are defined by iterating the previous one; however, unlike the operations before it, tetration is not an elementary function.

The parameter a is referred to as the base, while the parameter n may be referred to as the height. In the original definition of tetration, the height parameter must be a natural number; for instance, it would be illogical to say "three raised to itself negative five times" or "four raised to itself one half of a time." However, just as addition, multiplication, and exponentiation can be defined in ways that allow for extensions to real and complex numbers, several attempts have been made to generalize tetration to negative numbers, real numbers, and complex numbers. One such way for doing so is using a recursive definition for tetration; for any positive real

and non-negative integer
, we can define
recursively as:

The recursive definition is equivalent to repeated exponentiation for natural heights; however, this definition allows for extensions to the other heights such as

,
, and
as well – many of these extensions are areas of active research.

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https://www.mathisfunforum.com/profile.php?id=682 2022-03-07T13:24:05Z https://www.mathisfunforum.com/viewtopic.php?id=27219&action=new
<![CDATA[Some Special Numbers - Part 2.]]> a) 11 is the fifth prime and first palindromic multi-digit number in base 10.

b) 12 is the first sublime number.

In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number.

The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28.

There are only two known sublime numbers: 12 and

. The second of these has 76 decimal digits:

6,086,555,670,238,378,989,670,371,734,243,169,622,657,830,773,351,885,970,528,324,860,512,791,691,264.

c) We know 6 is the first perfect number : Sum of the factors whose proper factors sum to the number itself.

(1 + 2 + 3 = 6).

28 is the second perfect number.

496 is the third perfect number.

8128 is the fourth perfect number.

d) 17 is the sum of the first 4 prime numbers, and the only prime which is the sum of 4 consecutive primes.

e) 25 is the first centered square number besides 1 that is also a square number.

In elementary number theory, a centered square number is a centered figurate number that gives the number of dots in a square with a dot in the center and all other dots surrounding the center dot in successive square layers. That is, each centered square number equals the number of dots within a given city block distance of the center dot on a regular square lattice. While centered square numbers, like figurate numbers in general, have few if any direct practical applications, they are sometimes studied in recreational mathematics for their elegant geometric and arithmetic properties.

f) 30 is the smallest sphenic number.

In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers.

Definition : A sphenic number is a product pqr where p, q, and r are three distinct prime numbers. In other words, the sphenic numbers are the square-free 3-almost primes.

Examples : The smallest sphenic number is 30 = 2 × 3 × 5, the product of the smallest three primes. The first few sphenic numbers are

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, ...

g) 142857 is the smallest base 10 cyclic number.

A cyclic number is an integer in which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

142857 × 1 = 142857
142857 × 2 = 285714
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142.

h) 9814072356 is the largest perfect power that contains no repeated digits in base ten.

i) Pandigital number: In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 is a pandigital number in base 10. The first few pandigital base 10 numbers are given by :

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689.

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https://www.mathisfunforum.com/profile.php?id=682 2021-11-23T08:25:09Z https://www.mathisfunforum.com/viewtopic.php?id=26799&action=new
<![CDATA[x ° y = y ° x]]> Thank you zetafunc, I do appreciate the various solutions in case as they may come in handy

To be sure I understand I'm going to apply what you said first to the related case of

Taking logs of both sides we get:

Rearranging:

If we now make the y=x^t substitution:

And since y=x^t:

We now have a way to generate solutions. When I try to use this method to find those more complicated answers I began listing earlier, however, I still don't know how, other than by graphing it (which is very difficult for negative answers). For example, graphically it seems there must be some t the real part of which is around -2.56 for which

, yielding the solution to the original problem y=2, x=-0.767... . Is this what happens when you consider t over all the complex numbers? I can see now the solution space for positive t is rather simple (for both equations), but it's still tricky to assess when t is negative. I suppose I am still uninformed, and will come back in a little bit. ]]>
https://www.mathisfunforum.com/profile.php?id=212090 2021-10-23T22:11:23Z https://www.mathisfunforum.com/viewtopic.php?id=26703&action=new
<![CDATA[Self learning maths]]> You are in the right place. You can interact with many math enthusiasts! You can shared your knowledge with youngsters and teenagers.

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https://www.mathisfunforum.com/profile.php?id=682 2021-09-07T00:36:05Z https://www.mathisfunforum.com/viewtopic.php?id=25548&action=new
<![CDATA[Knuth's up-arrow notation]]> In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.

In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations. Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation. The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.

Various notations have been used to represent hyperoperations. One such notation is

. Another notation is
, an infix notation which is convenient for ASCII. The notation
is known as 'square bracket notation'.

Knuth's up-arrow notation

is an alternative notation. It is obtained by replacing
in the square bracket notation by
arrows.

For example:

the single arrow

represents exponentiation (iterated multiplication)

the double arrow
represents tetration (iterated exponentiation)

the triple arrow
represents pentation (iterated tetration)

The general definition of the up-arrow notation is as follows (for
:

Here,
stands for n arrows, so for example

.

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https://www.mathisfunforum.com/profile.php?id=682 2021-06-11T04:46:06Z https://www.mathisfunforum.com/viewtopic.php?id=26378&action=new
<![CDATA[e^x]]> As a teacher with many years of experience I know that the starting point in any lesson is to check that the pupils have understood the previous lesson.  From my point of view, there is no point spending my time adding the next steps if you haven't got a good grasp of what I have said so far.  I'm surprised you feel ready for the next step so soon after getting the link to the thread.

So I want something from you first:

Using a graph plotter, try a = 4 then 5 then 1 then 1/2  then 1/3.

Comment on what you are observing in these cases.  Make three conclusions about how the value of a affects the graph.

Then use the series expansion for e^x with x = 1 to compute e to 8 decimal places .

Post the steps in your working and your final result.  Pay particular attention to explaining how you know your answer is accurate to 8 dp (without just comparing with a published result).

Bob

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https://www.mathisfunforum.com/profile.php?id=67694 2021-05-15T19:45:24Z https://www.mathisfunforum.com/viewtopic.php?id=25550&action=new
<![CDATA[Would certain 3D objects be viewed as impossible in 4D?]]> Bob Bundy,

the impossible cube and the hypercube are different examples, right?

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https://www.mathisfunforum.com/profile.php?id=95904 2020-07-04T00:36:15Z https://www.mathisfunforum.com/viewtopic.php?id=25753&action=new
<![CDATA[How would I write an if statement in an equation?]]> Actually, piecewise functions are really just notation. Do you know how functions are usually viewed in modern mathematics?

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https://www.mathisfunforum.com/profile.php?id=95904 2020-07-01T00:56:02Z https://www.mathisfunforum.com/viewtopic.php?id=25745&action=new
<![CDATA[Imaginary number]]> Hi,

If r and θ are modulus and amplitude of a complex number,
then z=r(cosθ+isinθ).

Argument of Z and Amplitude of Z mean the same thing and are used interchangeably when we talk about complex numbers. When we plot the point of complex number on graph, and join it to the origin, the angle it makes with the x-axis is the argument or amplitude of complex number Z.

Amplitude.

Amplitude(or Argument) of a complex number:

Let z=x+iy  where x,y  are real,

and
; then the value of
for which the equations:

…(1) and

…(2)

are simultaneously satisfied is called the Argument(or Amplitude) of z  and is denoted by Arg z (or,

) .

Clearly, equations (1) and (2) are satisfied for infinite values of \theta ; any of these values of

is the value of Amp z . However, the unique value of
lying in the interval
and satisfying equations (1) and (2) is called the principal value of Arg z  and we denote this principal value by arg z  or amp z .Unless otherwise mentioned, by argument of a complex number we mean its principal value.

Since,

and
(where n=any integer), it follows that,

where
.

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https://www.mathisfunforum.com/profile.php?id=682 2020-04-14T01:51:50Z https://www.mathisfunforum.com/viewtopic.php?id=25576&action=new
<![CDATA[Limits]]> hi 666 bro

Thanks for that link to U Tube.  Now I understand your difficulty.

(1) Make sure your mic volume is set correctly.  Even on 100% volume I could barely hear the speaker.
(2) Plan your 'board' layout in advance so you don't have wobbly lines and have to rub out bits because they won't fit.
(3) Use different colours but only those that have a decent contrast with the black background.  Yellow is excellent; purple is very poor and barely readable.  Have printed text not handwritten notes so that we don't have to struggle reading your writing.

Sorry 666 bro but I felt I needed to get that off my chest.  No wonder you are struggling with this.  It's great that the Academy do this for free but they could learn a lot from MIF.  I suggest you look instead at this page:

https://www.mathsisfun.com/calculus/limits-formal.html

Compare the two and you'll see why I think MIF is such a brilliant resource.

Hope that helps,

Bob

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https://www.mathisfunforum.com/profile.php?id=67694 2020-03-29T12:54:10Z https://www.mathisfunforum.com/viewtopic.php?id=25566&action=new
<![CDATA[Sequences]]> Did you mean order in probability?

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https://www.mathisfunforum.com/profile.php?id=220123 2020-03-28T02:32:58Z https://www.mathisfunforum.com/viewtopic.php?id=25559&action=new