Mathematical Analysis

If you divide the number of words by the number of minutes, you'll get a rate of words per minute.

123

Factoring and Multiples

Mathematical Analysis

Here's what you do you find the smallest number and the larger number then subtract. For EXAMPLE: 12,18,22,24,25,32

32-12=20 so your RANGE is 20.

The possible x values of a function.

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Math and Arithmetic

Algebra

Mathematical Analysis

1000 thousands = 1 million , 1, 000, 000 1000 millions = 1 billion, 1,000,000,000 1000 billions = 1 trillion, 1,000,000,000,000 There are a thousand thousands in one million 1000 Proof is seen when multiplied together, 1000 * 1000 = 1,000,000 (one million).

There are: 1,000*1,000 = 1,000,000

150151152

Computers

Hard Disk Drives

Mathematical Analysis

Numbers

Zero Filling

- Zero filling is a way to overwrite all data on a hard drive. An analogy would be putting new asphalt over an old highway. This snip is from the Seagate Website: "(Quick) will write over the beginning of the drive which includes the critical partition information, eliminating all partitions and information on the drive including the Master boot record. This is useful if you have a drive that has a corrupted partition or that you wish to erase to reinstalll a fresh operating system and new data. (Full) will write over the entire data area of the drive. This is useful if a drive has bad sectors that cannot be fixed by the operating system. This will also erase all the data on the drive, but it will take several hours."

- The simplest way to think of zero filling is setting every single byte of a storage device (RAM, HDD, whatever it may be), to null (character code 0). This effectively cleans the device of any data. This full text is online at: http://www.seagate.com/support/kb/disc/faq/ata_llfmt_what.html

- Zero filling removes the data on a HD 100%. This is especially useful in preventing data being stolen from your HD. After doing a zero filling, (apparently) no tool can retrive data from your HD and everything is permanently deleted.

- Zero filling is a low level hard disk management technique. As it suggests, it "fills the hard disk with zeros". By that, I mean that it magnetically prepares the disk to be a fresh disk, consisting of only zeros, no ones.

- There is another common definition for 'Zero filling'. Typically, in software development you will have numeric entry fields where significant digits are important. If those digits are not entered you will 'zero fill' them to ensure the entire value is filled. For example, if you have a field which is six long and enter '123' then you will zero fill to get a value of '000123'.
- The U.S. Marines released a data spec a while back that required for the drive to be zeroed completely 7 times. it seems there are some ways to get data back even if it's been zeroed. It just costs ungodly amounts of money.

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Jobs & Education

Education

Mathematical Analysis

All things are sorted or categorized in this world. We need to know the proper equation to find the solution to which set theory is best for our purpose.

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Temperature

WikiAnswers Local

Mathematical Analysis

Numbers

80 is an integer. If you are asking for the Celsius equivalent of 80°F, then the answer is 26.555...°C

012

Math and Arithmetic

Mathematical Analysis

Abstract Algebra

In math, symbolic logic is simply expressing a mathematically logical statement through the use of symbols. For instance, one could always write down the phrase, "one plus one equals two," but using symbolic logic, that statement can be expressed much more succinctly as 1 + 1 = 2.

A better example is:

The indefinite integral of one divided by the quantity one minus the square of x with respect to x is equal to one half multiplied by the natural logarithm of the quotient of the quantities one plus x and one minus x with the constant of integration added to this result

Symbolically written, that statement is expressed as:

∫ [1/(1 - x2)] dx = ½ ln[(1 + x)/(1 - x)] + C,

which is a whole heck of a lot easier to write!

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Math and Arithmetic

Algebra

Mathematical Analysis

W-4=11

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Math and Arithmetic

Mathematical Analysis

Abstract Algebra

There are two types of mathematical axioms: logical and non-logical.

Logical axioms are the "self-evident," unprovable, mathematical statements which are held to be universally true across all disciplines of math. The axiomatic system known as ZFC has great examples of logical axioms. I added a related link about ZFC if you'd like to learn more.

Non-logical axioms, on the other hand, are the axioms that are specific to a particular branch of mathematics, like arithmetic, propositional calculus, and group theory. I added links to those as well.

285286287

Math and Arithmetic

Algebra

Mathematical Analysis

A part to part ratio considers and compares the probabilities of two different outcomes happening. For example, a part to part ratio could be HEADS to TAILS on a fair coin. The ratio is 1/2 to 1/2. Simplified, this is 1 to 1.

However, a part to whole ratio compares the probability of one outcome occurring to the total possibilities possible. Consider the example of a fair coin. A part to whole ratio would be HEADS to ALL EVENTS POSSIBLE or 1/2 to (1/2 + 1/2) or 1/2 to 1 or simplified, 1 to 2

456

Statistics

Trigonometry

Mathematical Analysis

distance and angle

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Math and Arithmetic

Statistics

Mathematical Analysis

No, it is a discrete variable. Since there are no fractional people, a count of people can only be a positive integer. For a variable to be continuous, it must be able to take on *any* real value in a domain. So, if populations could take on any real value, including rational and irrational ones, between 1 and say 10^12, including ones like 6.1385391..., then population would be a continuous variable.

269270271

Science

Math and Arithmetic

Physics

Mathematical Analysis

Vectors have a lot of applications in physics and engineering. Velocity, acceleration and forces are vectors.

For example, you can use vectors to determine the total distance travelled by a boat travelling across a river at 25 miles/hour while the river flows perpendicular to the boat at 10 miles/hour.

There are loads of other applications. Vectors can be used to predict the movement of planets in their orbits, and are useful for 3D geometry in general. Vectors are also essential in electromagnetics, where they are used to represent electric and magnetic fields and their relationships. Maxwell's equations are equations which use vectors. The Poynting vector is used to calculate the flow of electromagnetic energy through space, for example by radio waves or light.

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Math and Arithmetic

Geometry

Probability

Mathematical Analysis

An open circle means that the value that it refers to is not part of the valid solution. A close circle indicates that it is.

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Electronics Engineering

Electrical Engineering

Mathematical Analysis

Ajw *sa(x)

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Math and Arithmetic

Mathematical Analysis

Numbers

Mathematical Constants

In ordinary mathematics, you may not divide by zero. It is considered undefined.

Consider the two situations:

For the inverse of multiplication 0/0 = a, a could be any number to satisfy a x 0 = 0. At the same time, division of any nonzero number, a/0 = b, there is no number a such that b x 0 = a.

---

In nearly every known algebraic structure, 0/0 is an undefinable term. This means that, based on the rules that govern most of the mathematical systems we use, there isn't just one, single, definable value for the term 0/0, and believe it or not, the reason for this isn't because we're dividing by zero, it's because the division relation is defined by another relation, multiplication. You see, when we talk about "divide," what we really mean is "multiply by the inverse." For example, x/y actually means, x*y-1 where y-1 is the inverse of y. The inverse of a number is defined to be the number which, when multiplied by the original number, equals one; e.g. x*x-1 = 1. Now, in the algebraic structures we're all familiar with, any number multiplied by zero is defined to be equal to zero; e.g. 0*x = 0. So, using these definitions, what does dividing by zero, which actually means multiplying by the inverse of zero, equal? In other words, x*0-1 = ? Well, to isolate x, you would need to cancel out 0-1, but how? As anyone who's taken any sort of algebra knows, the method of isolation in these cases would be to multiply 0-1 by 0 because, as stated above, x*x-1 = 1, therefore 0*0-1 = 1. But wait, didn't I also just say that 0*x = 0? That would mean that 0*0-1 = 0, which would mean that 0 = 1. That, my friends, is called a contradiction. Zero does not equal one; therefore the term 0-1 can't be defined.

This answer may seem unsatisfactory to some people. There's got to be a way to work around this pesky contradiction, right? Actually, there is! In the branch of mathematics called abstract algebra, there exists an algebraic structure called a wheelwhich is required to have division defined everywhere within it. Therefore, in this particular algebraic structure, 0/0 must exist or else the structure isn't a wheel.

But wait, 0/0 is undefined, right? How could you ever satisfy this requirement for a wheel then?

That's easy; all you have to do is define it! Specifically, you give this quantity, 0/0, some specific algebraic properties, and then, if it ever comes up in an equation, you manipulate it within the equation using the properties you've given it. Isn't that convenient?!

"Preposterous!" you may say. "You can't simply make something up which has no tangible or rational analogue, that's cheating!" Well my dear skeptic, may I direct your attention to the following little marvel, √(-1), otherwise known as "the imaginary number," or i. That's right, I said imaginary, as in, "doesn't exist." You see, nothing multiplied by itself in our nice little world of mathematical rationality can possibly be a negative number. Unless, of course, you define something to be as such. Then...Presto! The absurd is now reality!

Let's talk about imaginary numbers for a moment. Our newly defined yet still rather imaginary friend, i, was apparently not content on simply having a nice, comfy little existence within the realm of obscure mathematics, oh no no no. It decided to defy logic and become a fairly common number; popping up all over the place, even in (you're going to love this) actual, real-life applications. For example, anyone who's ever done some form of electromagnetic wave analysis, through the fields of engineering, physics, etc., LOVES i and will gladly bow down and kiss its feet upon command (God bless ei(ωt-kr)). Why? Because of the very thoughtful relation that it's given to us known as "Euler's formula:" eiθ = cos(θ) + isin(θ). Step back a minute and look at that. The irrational, real number, e (2.71828...) exponentiated to the product of a real number, θ, and the imaginary number, i, is equal to a simple trigonometric expression involving two basic functions. In fact (you may want to sit down for this), if the value for θ happens to be π (3.14159...), another irrational, real number mind you, the trigonometric expression on the right hand side of Euler's formula reduces to exactly -1. Let's write that out: eiπ = -1. We call that "Euler's identity," although it should really be called, "THE MOST INCREDIBLE MATHEMATICAL EXPRESSION, EVER!"

But enough about i, let's get back to our newest friend, 0/0. As stated earlier, the problem with 0/0 isn't the fact that we're dividing by zero, it's the fact that the division relation is defined by multiplication. Well, how do we fix that? Simple! Change the definition of divide! Instead of x/y = x*y-1, it's now going to equal x*/y, where "/" is defined as a unary operation analogous to the reciprocal operation.

OK, another quick aside. A unary operator is an operator that only needs one input to work. For example, you only need one number to perform the operation of negation. For instance, negating the number 1 is simply -1. This is opposed to a binaryoperator. Binary operators include many of the guys we're all familiar with; like addition, multiplication, subtraction, etc. To make this clearer, consider the addition operation. It would make no sense to write 1 +. You need another number after the "+" to satisfy the operation; 1 + 2, for example, hence the term binary.

So, with our trusty new unary operator "/" in hand, we're going to look at the number 0/0 again. 0/0 is no longer defined as 0*0-1 like it was before. Now, it's defined as 0*/0, and in our world, not only does /0 ≠ 0-1, but 0*x doesn't have to equal 0 either. Isn't abstraction fun?! Ok, so 0/0 is officially defined, now let's give it some properties!

How about, x + 0/0 = 0/0 and x*0/0 = 0/0. Awesome! Why not go ahead and make a more general rule as well: (x + 0y)z = xz + 0y. OK! Well, we're certainly off to a good start, I'd say. I'll leave the complete derivation of the algebraic structure known as the wheel to the experts, please see the corresponding link below.

I'll end this answer with a final note for those who think that this entire concept of "defining the undefined" is ridiculous. Consider the following sets of numbers: The prime numbers, P; the set of all real numbers with exactly two natural number factors.

The natural numbers, N; the set of all integers greater than or equal to 0.

The integers, Z; the set of all real numbers without remainders or decimals.

The rational numbers, Q; the set of all real numbers that can be expressed as an integer divided by a non-zero integer.

The irrational numbers, I; the set of all real numbers that aren't rational.

Now consider this:

The imaginary number, i, is undefined in I.

The ratio pi, or π (3.14159...), is undefined in Q.

The common fraction 1/2 is undefined in Z.

All of the negative numbers, including -1, are undefined in N.

The number 4 is undefined in P.

Yet, these "undefined" numbers are hardly mysterious to us. We just broadened our definition of definable to include the "undefined" ones, and life became good again. 0/0 is not quite, but nearly, the same idea.

--------------------------------------------------------

I once asked one of my professor lecturers at University this and his answer was any value you want (or need).

0/0 is used as a limit in Calculus.

Consider any curve y = f(x)

Take a point (x, f(x)) on that curve.

The slope of that point is the slope of the tangent at that point.

The slope of the tangent is close to the slope of a small chord between the point (x, y) = (x, f(x)) and a point a small distance h away (x+h, f(x+h)), which can be found by: m = (f(x+h) - f(x))/((x+h) - x) = (f(x+h) - f(x))/h

The smaller the value of h, the closer the chord is to the tangent and the closer the slope of the chord is to the slope of the tangent and thus the slope of the curve at that point.

As h tends towards 0, f(x+h) tends towards f(x) and the expression m = (f(x+h) - f(x))/h tends towards 0/0.

In other words, 0/0 is the limit of (f(x+h) - f(x))/h as h tends towards 0.

But as this chord tends towards the tangent at the point (x, f(x)) on the curve y = f(x), 0/0 must be the slope of the tangent.

Clearly not every point of a non-linear curve has the same slope, thus 0/0 is any value you want (or need).

As the chord tends towards having zero length (when h = 0), (f(x+h) - f(x))/h will tend towards a constant value, a limit, which is the slope of the tangent.

The "trick" that calculus uses is that as h never reaches 0 but tends towards 0 it is possible to divide by h, and then see what happens when h becomes 0, ie when the original expression became 0/0, since (f(x+h) - f(x))/h = (f(x+0) - f(x))/0 = (f(x) - f(x))/0 = 0/0 when h = 0.

For example, take the curve y = x³ - 2x² + 5x + 3; what is the value of the slope of that line?

slope = lim{h→0} (f(x+h) = f(x))/h

= lim{h→0} ((x+h)³ - 2(x+h)² + 5(x+h) + 3 - (x³ - 2x² + 5x + 3))/h

Expanding the brackets:

= lim{h→0} (x³ + 3x²h +3xh² + h³ - 2x² - 4xh - 2h² + 5x + 5h + 3 - x³ + 2x² - 5x - 3)/h

Simplifying:

= lim{h→0} (3x²h +3xh² + h³ - 4xh - 2h² + 5)/h

Since h ≠ 0, it is possible to divide by h:

= lim{h→0} 3x² +3xh + h² - 4x - 2h + 5

Now the limit can be found by letting h = 0:

= 3x² - 4x + 5

Thus the slope of y = x³ - 2x² + 5x + 3 is given by m = 3x² - 4x + 5 at any value for x.

This value m, which is normally written as f'(x) is the first derivative of f(x), also written as dy/dx.

The slope of any line y = f(x) is given by y = f'(x).

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Mathematical Analysis

Mathematicians

Assuming you mean "the calculus of variations," all the big names were involved in its development; Bernoulli, Euler, Lagrange, Legendre, Cauchy, Clebsch, Weierstrass, Hilbert, and Lebesgue to name a few.

186187188

Math and Arithmetic

Mathematical Analysis

Numbers

The definite integral of any function identically equal to zero between any two points is zero.

Integral is the area under the graph of the given function. Sometimes the terms "integral" or "indefinite integral" are used to refer to the general antiderivative of a function, especially in many textbooks. In this case, the indefinite integral is equal to an arbitrary constant, and it is important to distinguish between these two cases.

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Inventions

Electrical Engineering

Industries and Professions

Mathematical Analysis

Mathematicians

AC circuit analysis, for one.

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Geometry

Numerical Analysis and Simulation

Mathematical Analysis

- a ikosikaicentagon

158159160

Geometry

Mechanical Engineering

Mathematical Analysis

to calculate the number of possible triangles within a nonagon, one must use combinatorics. Understand that at first you have to pick the first vertex from 9 points, then the second from eight other possible points, and the third from 7 possible points. And finally that by this process any three vertices could be picked in 6 different orders. triangles possible are: 9*8*7/6 = 84 possible triangles in a nonagon

91011

Math and Arithmetic

Mathematical Analysis

1 x 231, 3 x 77, 21 x 11, 33 x 7.

Add the digits and you get 6, which means it is divisible by 3. (3 x 77) 77 has two prime factors, 7 and 11. So you can multiply any combination of the factors to get 231. 3C2 (3 choose 2) in Pascal's triangle is 3, which means there are three possible combinations. Oh, yeah, and you can always just do 1 x 231.

333435

Math and Arithmetic

Mathematical Analysis

Linear Algebra

When performing the cross product of two vectors (vector A and vector B), one of the properites of the resultant vector C is that it is perpendicular to both vectors A & B. In two dimensional space, this is not possible, because the resultant vector will be perpendicular to the plane that A & B reside in. Using the (i,j,k) unit vector notation, you could add a 0*k to each vector when doing the cross product, and the resultant vector will have zeros for the i & jcomponents, and only have k components.

Two vectors define a plane, and their cross product is always a vector along the normal to that plane, so the three vectors cannot lie in a 2D space which is a plane.

150151152

Math and Arithmetic

Mathematical Analysis

An idealised line.

149150151

Mathematical Analysis

There is > and + is all i know

Subtraction: 50 decreased by 10 is the same as 50 - 10.

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