In mathematics, two quantities are in the silver ratio (or silver mean) if the ratio of the smaller of those two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity. This defines the silver ratio as an irrational mathematical constant, whose value of one plus the square root of 2 is approximately 2.4142135623. Its name is an allusion to the golden ratio; analogously to the way the golden ratio is the limiting ratio of consecutive Fibonacci numbers, the silver ratio is the limiting ratio of consecutive Pell numbers. The silver ratio is denoted by δS.

Mathematicians have studied the silver ratio since the time of the Greeks (although perhaps without giving a special name until recently) because of its connections to the square root of 2, its convergents, square triangular numbers, Pell numbers, octagons and the like.

or equivalently,

The silver ratio can also be defined by the simple continued fraction [2; 2, 2, 2, ...]:

are ratios of consecutive Pell numbers. These fractions provide accurate rational approximations of the silver ratio, analogous to the approximation of the golden ratio by ratios of consecutive Fibonacci numbers.

The silver rectangle is connected to the regular octagon.

]]>The best -

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

]]>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

]]>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.**Addition**

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.]]>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.

]]>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. ]]>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.]]>

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

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

]]>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.

See this link:

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

Teacher's advice to Khan academy:

(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

]]>