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Proofs
Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.
1,294 Questions
How can I prove an angle bisector?
It depends on what is given.
In general, one half of the bisected angle is proven to congruent to the other half. By the Definition of an Angle Bisector, the bisected angle can be proven bisected.
---- To show that two angles are congruent:
One way to prove the two angles congruent is to show that their measures are equal. This can be done if there are numbers on the diagram. Use the Protractor Postulate or the Angle Addition Postulate to find the smaller angles' measures, if they are not directly marked. Then use the Definition of Congruent Angles to prove them congruent.
Given that the smaller angles correspond on a congruent or similar pair of figures in that plane and form an angle bisector, the Corresponding Parts of Congruent Figures Postulate or Corresponding Parts of Simlar Figures Postulate may be used.
What combination of numbers always results in an odd number?
Multiply two odd numbers Add an even and an odd Subtract an odd
and an even
What are example of philosophizing?
Examples of philosophizing are readily found within the texts written by acknowledged philosophers from past and present: such texts offer the 'philosophizings' of some of the most active minds in human history. Additionally, examples of philosophizing can be found in daily life among less famous but (perhaps) no less philosophical minds: every time that ordinary persons ask an important question in a serious way, then proceed to seek an answer through a rational (or, mind-driven) process, philosophizing is taking place.
What is the difference between a straight line and a line?
A straight line is the shortest distance between two points, a line is the delineation of a connection between two or more points.
log10 0 is actually undefined. Think about it like this:
If loba b = y then we know that ay = b
This means that log10 0 = y translates to 10y = 0
But as you know, 10y is always greater than zero.
Therefore 10y = 0 is undefined.
Therefore log10 0 = y is undefined.
The line on surface of earth joining the points where h has same value is what?
The answer depends on what h is supposed to represent.
How do you prove Euler's formula?
Euler's formula states that eix = cos(x) + isin(x) where i is the imaginary number and x is any real number.
First, we get the power series of eix using the formula:
ez = Σ∞n=o zn/n! where z = ix. That gives us:
1 + ix + (ix)2/2! + (ix)3/3! + (ix)4/4! + (ix)5/5! + (ix)6/6! + (ix)7/7! + (ix)8/8! + ...
which from the properties of i equals:
1 + ix - x2/2! - ix3/3! + x4/4! + ix5/5! - x6/6! - ix7/7! + x8/8! + ...
which equals:
(1 - x2/2! + x4/4! - x6/6! + x8/8! - ...) + i(x - x3/3! + x5/5! - x7/7! + ...).
These two expressions are equivalent to the Taylor series of cos(x) and sin(x). So, plugging those functions into the expression gives cos(x) + isin(x).
Q.E.D.
Of course, were you to make x = п in Euler's formula, you'd get Euler's identity:
eiπ = -1
It depends on your definitionThe question presumes that someone has already given a definition of the exponential function for complex numbers. But has this definition been given? If so, what is this official definition?The answer above assumes that the exponential function is defined, for all complex numbers, by its power series. (Or, at least, that someone else has already proved that the power series definition is equivalent to whatever we're taking to be the official definition).
So the answer depends critically on what the definition of the exponential function for complex numbers is.
Suppose you know everything about the real numbers, and you're trying to build up a theory of complex numbers. Most people probably view it this way: Mathematical objects such as complex numbers are out there somewhere, and we have to find them and work out their properties. But mathematicians look at it slightly differently. If we can construct a thing which has all the properties we feel the field of complex numbers should have, then for all practical purposes the field of complex numbers exists. If we can define a function on that field which has all the properties we think exponentiation on the complex numbers should have, then that's as good as proving that complex numbers have exponentials. Mathematicians are comfortable with weird things like complex numbers, not because they have proved that they exist as such, but because they have proved that their existence is consistent with everything else. (Unless everything else is inconsistent, which would be a real pain but is very unlikely.)
So how do we construct this exponential function? One approach would be simply to define it by exp(x+iy) = ex(cos(y)+i.sin(y)). Then the answer to this question would be trivial: exp(iy) = exp(0+iy) = e0(cos(y)+i.sin(y)) = cos(y)+.sin(y). But you'd still have some work to do to prove things like exp(z+w) = exp(z).exp(w). Alternatively, you could define the exponential function by its power series. (There's a theorem that lets you calculate the radius of convergence, and that tells us the radius is infinite, i.e. the power series works everywhere.) Or maybe you could try something like proving that the equation dw/dz = w has a unique solution up to a multiplicative constant, and defining the exponential function to be the solution which satisfies w=1 at z=0.
Let z = cos(x) + i*sin(x)
dz/dx = -sin(x) + i*cos(x)
= i*(i*sin(x) + cos(x))
= i*(cos(x) + i*sin(x))
= i*(z)
therefore dz/dx = iz
(1/z)dz = i dx
INT((1/z)dz = INT(i dx)
ln(z) = ix+c
z = e(ix+c)
Substituting x=0, we get cos(0) + i*sin(0) = e(0i+c)
1+0 = e(0+c)
e(0+c) = 1
therefore c=0
z = eix
eix = cos(x) + i*sin(x)
Substituting x=pi, we get e(i*pi) = -1 + 0
e(i*pi) = -1
e(i*pi) + 1 = 0
qED
What are the Details about the history of number zero?
There are two different things to consider:
(*) a place-holder for an empty column in a positional number system
(*) zero as a number in its own right.
Our number system is positional, so in the number 33, the first 3 means three lots of ten, and the second 3 means three units. If we didn't have 0, we would have trouble distinguishing between thirty-three and three hundred and three. We could write three hundred and three like this:
3 3 , leaving a space for the tens column.
This is prone to mistakes, so we could instead write three hundred and three like this:
3*3 , using * to indicate that the tens column is empty.
Something like using space and * as place-holders was done by the Babylonians about 3500 years ago. This is not the same thing as treating 0 (zero) as a number in its own right. That seems to have been first done by the Indian mathematician Brahmagupta, in 628 AD.
Let L be the length and W be the width of the rectangle.
Given:
2(W+L) = 98 -- (1)
L = 2W + 4 ---- (2)
Solution:
Rearranging (2), we have 2W = L - 4 --- (3)
Substituting (3) into (1). We get (L-4) + 2L = 98.
L = 34 ---------- (4)
Substituting (4) into (2), we have 34 = 2W + 4.
W = 15.
How do you find the equation of a tangent line?
In order to find the equation of a tangent line you must take the derivative of the original equation and then find the points that it passes through.
What additional information is needed to prove abc equals aed by HL THEROM?
The answer depends on what information you already have. Without that knowledge, you cannot even begin to guess what is additional.
How do you prove that if a real sequence is bounded and monotone it converges?
We prove that if an increasing sequence {an} is bounded above, then it is convergent and the limit is the sup {an }
Now we use the least upper bound property of real numbers to say that sup {an } exists and we call it something, say S. We can say this because sup {an } is not empty and by our assumption is it bounded above so it has a LUB.
Now for all natural numbers N we look at aN such that for all E, or epsilon greater than 0, we have aN > S-epsilon. This must be true, because if it were not the that number would be an upper bound which contradicts that S is the least upper bound.
Now since {an} is increasing for all n greater than N we have |S-an|
How do we prove that a finite group G of order p prime is cyclic using Lagrange?
Lagrange theorem states that the order of any subgroup of a group G must divide order of the group G. If order p of the group G is prime the only divisors are 1 and p, therefore the only subgroups of G are {e} and G itself. Take any a not equal e. Then the set of all integer powers of a is by definition a cyclic subgroup of G, but the only subgroup of G with more then 1 element is G itself, therefore G is cyclic. QED.
Do a triangle with the sides lengths of 16 30 and 35 make a right triangle?
No.
The Pythagorean theorem states that a triangle is a right triangle if and only if a2+b2=c2, where a, b, and c are the lengths of the sides of the triangle.
162+302 = 256+900 = 1156
352 = 1225
Since 1156 does not equal 1156, this is not a right triangle.
Let A be any set that has at least 2 elements and let every element in set A maps to itself except for 2 elements that do not map to each other. i.e.
A = 1, 2, 3, ... , n
1 --> 2
2 --> 1
3 --> 3
4 --> 4
...
n --> n
What is formula of perimeter of semi circle?
The perimeter of a circle has the formula 2πr. Therefore the length of a semi circle is πr. But the perimeter of a semi circle also includes the diameter which is 2r.
Therefore the perimeter of a semi circle = πr + 2r = r(π + 2)
The word zillion is a fictitious amount, and is used to express an unspecified but extraordinarily large number.
The word reflects the English number names of the -illionform, and the Z is added to make it look like one of those really large numbers.
A zillion is a very large indefinite number.
