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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 do you prove that the volume of a sphere is equal to the volume of a cone?
The volume of a cone is 1/3(h)(pi)(r2), where h is the height of the cone, pi is 3.1415 and r is the radius of the circle that forms the bottom.
The volume of sphere is 4/3(pi)(r2) where pi is 3.1415 and r is the radius of the sphere.
The (r2) means radius squared. If you put in the values of r for each and the value of h for the cone and solve the two equations, and the answers are the same, the volumes are the same. We can set the expression for the volume of a cone equal to the expression for the volume of a sphere. If, when we plug in the variables, they are equal, the volumes will be equal. Vcone = Vsphere 1/3 (h) (pi) (rc2) = 4/3 (pi) (rs2)
Is binary numbers is the internal language of computer?
A computer only understand binary, which is 0 as "off" and 1 as "on."
How can you say that if areas of two triangles are equal then they are congruent?
In order for a triangle to be congruent the two triangles have to be the same shape and size, thus they are congruent if they can be moved into an isometry or any other combination.
But you're asking how a question which has two possibilities.
Assuming that you have two triangles whose sides are equivalent which makes the areas equal to each other then you can state the side-side-side rule which is if the three sides of one triangle is equivalent to the other three sides of the other triangle then they are congruent.
But if you have an angle present in the triangles you could argue the angle angle side rule,
but if the angles are joint you would argue the angle side angle.
But if one triangle has one degree and the other one has a different degree then they will not be congruent.
How many times can you recycle paper?
it depends on the weight of the egg after the volcano had a child.
Are absolute value and additive inverse equal why?
No. This is because absolute values are always positive.
For example: |2|=2 absolute value
Additive inverse means the opposite sign of that number so 2's additive inverse is -2. But sometimes if the number is -2 then the additive inverse equals the absolute value. therefore the answer is sometimes
What is the formula for radius?
A radius is half of the diameter of a circle, or the distance from one endpoint to the center.
How do you prove the Baire Category Theorem?
The Baire Category Theorem is, in my opinion, one of the most incredible, influential, and important results from any field of mathematics, let alone topology. It is known as an existence theorem because it provides the necessary conditions to prove that certain things must exist, even if there aren't any examples of them that can be shown. The theorem was proved by René-Louis Baire in 1899 and is a necessary result to prove, amongst other things, the uniform boundedness principle and the open mapping theorem (two of the three most fundamental results from functional analysis), the real numbers being uncountable, and the existence of continuous, yet nowhere differentiable, functions from R to R.
The proof is quite long and involves some pretty advanced math, so to help with the reader's comprehension there is a list of symbols and their meanings at the end of this proof. Also, I've added many related links with definitions and explanations of the terms used in this proof.
The Baire Category Theorem:
If B, D is a nonempty, complete metric space, then the following two statements hold:
1) If B is formulated as the union of countably-many subsets, C1, C2, …, Cp, then at least one of the Cp is somewhere dense.
2) If A1, A2, …, Ap are countably-many, dense, open subsets of B, then ∩pAp is dense in B, i.e. Cl(∩pAp) = B
Proof:
1) If the first statement is false, then there is a countable family {Cp}, p Є P, of subsets of B such that B = ∪pCp, but (Cl Cp)o = Ø for each p Є P. Therefore, for each p, Cl Cp≠ B. Select b1 Є B - Cl C1. There is a positive number m1 < 1, since B - Cl C1 is open, such that N(b1, m1) ⊂ B - Cl C1. Now we set G1 = N(b1, m1/2). Then Cl G1 ⊂ N(b1, m1); hence Cl G1 ∩ Cl C1 = Ø.
Since G1 is a nonempty, open subset of B, that means G1 ⊄ Cl C2. So, choose a b2 Є G1- Cl C2. Since G1 - Cl C2 is open, there is an m2 > 0 such that N(b2, m2) ⊂ G1 - Cl C2. This time we'll require m2 < 1/2 and then set G2 = N(b2, m2/2). Then G2 ⊂ G1 and Cl G2 ∩ Cl C2 = Ø.
If we continue on like this, requiring m3 < 1/3, m4 < 1/4, etc., we'll obtain a decreasing sequence of mp-neighborhoods, G1 ⊃ G2 ⊃ G3 ⊃ … ⊃ Gp ⊃ … such that Cl Gp ∩ Cl Cp = Ø and mp < 1/p. Then Cl G1 ⊃ Cl G2 ⊃ Cl G3 ⊃ … ⊃ Cl Gp ⊃ … and d(Gp) --> 0.
I'm going to use a result from another theorem in topology, not proven here, which says that if G1 ⊃ G2 ⊃ G3 ⊃ … ⊃ Gp ⊃ … , d(Gp) --> 0, and ∩pGp ≠ Ø, then the metric space B, D is complete. Therefore, ∩p Cl Gp ≠ Ø. So, if we pick a g Є ∩pGp, then g Є Cp for some p, since ∪pCp = B. However, that would imply that g Є Cl Cp ∩ Cl Gp which is impossible because Cl Cp and Cl Gp are disjoint. So, for 1), Q.E.D.
2) To start, we're going to suppose that {Ap}, p Є P, is a countable family of dense, open subsets of B. To prove that ∩pAp is dense, all that we need to prove is that every neighborhood of any element of B meets ∩pAp. In other words, for any selected g Є Band any m > 0, we'll show that N(g, m) ∩ (∩pAp) ≠ Ø.
If we set T = Cl N(g, m/2), then T ⊂ N(g, m). Now we'll show that T ∩ (∩pAp) ≠ Ø. We know that T is a subspace of the closed metric space, B, D, and that Titself is closed. So, using an earlier theorem that won't be proved here, T is a complete metric space. If we set Gp = T - Ap which is equal to Tp ∩ (B - Ap), we see that the intersection of two closed subsets of B, Gp is closed in both B and T.
Now suppose Gp is somewhere dense. Then there is an element t Є T and a number q > 0 such that N(t, q) ∩ T ⊂ Cl Gp ∩ T = Gp. Therefore, N(t, q) ∩ (T - Gp) = Ø. We can see that t Є T = Cl N(g, m/2). Therefore N(t, q) meets N(g, m/2) at some point z. We then choose q' > 0 such that N(z, q') ⊂ N(t, q) ∩ N(g, m/2). However, since Ap is dense, N(z, q') intersects Ap at a point we'll call z' . Well, then it must be that z' Є N(t, q) ∩ T ⊂ Gp. But, Gp= T - Ap, hence z' Є T - Ap. That implies then that z' ∉ Ap which is a contradiction. Therefore Gp must be nowhere dense in T.
So, by the first statement of the theorem, 1), which we already proved, T ≠ ∪pGp. Thus, there is n element s Є T - ∪pGp. Therefore, since Gp = T - Ap, then s Є T ∩ (∩pAp) and so T∩ (∩pAp) ≠ Ø meaning N(g, m) ∩ (∩pAp) ≠ Ø.
Q.E.D.
List of symbols:
R - The set of real numbers, including rational, irrational, positive, and negative numbers, as well as 0. Not including complex numbers having an imaginary part other than 0i, where i is the imaginary number √(-1).
B, D - The metric space of set B with metric D.
∩pAp - The intersection of all of the subsets A1, A2, …, Ap.
Cl - The closure of whatever set is written after it.
p Є P - p is an element of the set P.
P - The set of all positive integers, not including 0. This set is often referred to as the set of natural numbers and is labeled N, but since at times the natural numbers are said to include 0, I've labeled this set P to avoid ambiguity. Not to mention, I've used N within my label for neighborhood.
∪pCp - The union of all of the of the subsets C1, C2, …, Cp.
( )o - The interior of whatever set is in the parentheses.
Ø - The empty set; i.e. the set with nothing in it.
N(b1, m1) - The neighborhood of point b1 within distance m1.
⊂ - … is a subset of …
⊃ - … is a superset of …
d( ) - The diameter of whatever set is in the parentheses.
--> 0 - The limit of whatever comes before the symbol "-->" goes to 0.
What type of electron orbital s p d f is designated by n equals to 2 l equals to 0and m equals to 0?
S orbital
If the diagonals of a parallelogram are congruent the parallelogram may or may not be a rectangle?
False
Are there any apps that solve geometric proofs for you?
Photomath reads and solves mathematical problems instantly by using the camera of your mobile device.
An antiderivative, F, is normally defined as the indefinite integral of a function f. F is differentiable and its derivative is f.
If you do not assume that f is continuous or even integrable, then your definition of antiderivative is required.
What is the multicative identity of -3?
-3 does not have a multiplicative identity in the set of real numbers.
Let G be a cyclic group of order 8 then how many of the elements of G are generators of this group?
Four of them.
How can you prove that the square root of two is transcendental?
You can't, because it isn't. The square root of 2 is irrational, but that doesn't make it transcendental. The square root of any positive integer is ALGEBRAIC - and transcendental means "not algebraic".In this case, the square root of 2 is a root of the polynomial equation x squared - 2 = 0; therefore it is algebraic.
If the cuboid has sides of length s units then its volume is s^3 = s*s*s cubic units.
Zillion is not a number,
along with Ba-zillion and Ga-zillion (which would be cool though)
Zillion is really used as "slang" indicating an infinite or very large amount.
A extremely large yet unspecified number mostly used in a hypothetical sense.
So to sum it up
NO, ZILLION is not a number.
Brief history of trigonometry?
Trigonometry was probably developed for use in sailing as a navigation method used with astronomy.[2] The origins of trigonometry can be traced to the civilizations of ancient Egypt, Mesopotamia and the Indus Valley, more than 4000 years ago.[citation needed] The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian's base sixty system of numeration. The Sulba Sutras written in India, between 800 BC and 500 BC, correctly computes the sine of (=45°) as in a procedure for "circling the square" (i.e., constructing the inscribed circle).[citation needed]
The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus[1] circa 150 BC, who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations circa 100 AD.
The ancient Sinhalese in Sri Lanka, when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 BC.[3]
The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton-Stirling interpolation formula.
In the 10th century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, e.g. sin (a + b), and discovered the sine formula for spherical geometry:
Also in the late 10th and early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula
.
Indian mathematicians were the pioneers of variable computations algebra for use in astronomical calculations along with trigonometry. Lagadha (circa 1350-1200 BC) is the first person thought to have used geometry and trigonometry for astronomy, in his Vedanga Jyotisha.
Persian mathematician Omar Khayyám (1048-1131) combined trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. Khayyam solved the cubic equation x3 + 200x = 20x2 + 2000 and found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables.
Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry.
The 13th century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.
In the 14th century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur) produced tables of trigonometric functions as part of their studies of astronomy.
The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry".
elementary proof
