0
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
What best way to teach mathematics so that the child will not be afraid?
What best way to teach mathematics
total use of food cost divided by net sales is food cost percentage
No. You can know all three angles of both and all you can say is that the triangles are similar.
Or with any pair of congruent sides you can have an acute angle between them or an obtuse angle.
Not possible to answer - a litre is a measurement of volume, while a kilogram is a measurement of weight!
Prove that a group of order three is abelian?
By LaGrange's Thm., the order of an element of a group must divide the order of the group. Since 3 is prime, up to isomorphism, the only group of order three is {1,x,x^2} where x^3=1. Note that this is a finite cyclic group. Since all cyclic groups are abelian, because they can be modeled by addition mod an integer, the group of order 3 is abelian.
Is the square root of 14 an irrational number?
Yes, here's the proof.
Let's start out with the basic inequality 81 < 83 < 100.
Now, we'll take the square root of this inequality:
9 < √83 < 10.
If you subtract all numbers by 9, you get:
0 < √83 - 9 < 1.
If √83 is rational, then it can be expressed as a fraction of two integers, m/n. This next part is the only remotely tricky part of this proof, so pay attention. We're going to assume that m/n is in its most reduced form; i.e., that the value for n is the smallest it can be and still be able to represent √83. Therefore, √83n must be an integer, and n must be the smallest multiple of √83 to make this true. If you don't understand this part, read it again, because this is the heart of the proof.
Now, we're going to multiply √83n by (√83 - 9). This gives 83n - 9√83n. Well, 83n is an integer, and, as we explained above, √83n is also an integer, so 9√83n is an integer too; therefore, 83n - 9√83n is an integer as well. We're going to rearrange this expression to (√83n - 9n)√83 and then set the term (√83n - 9n) equal to p, for simplicity. This gives us the expression √83p, which is equal to 83n - 9√83n, and is an integer.
Remember, from above, that 0 < √83 - 9 < 1.
If we multiply this inequality by n, we get 0 < √83n - 9n < n, or, from what we defined above, 0 < p < n. This means that p < n and thus √83p < √83n. We've already determined that both √83p and √83n are integers, but recall that we said n was the smallest multiple of √83 to yield an integer value. Thus, √83p < √83n is a contradiction; therefore √83 can't be rational and so must be irrational.
Q.E.D.
This proof uses modular arithmetic. If you are unfamiliar with this, the basic principle is that if we have integers a, b, and a nonzero integer c, then a = b (mod c) if a/c and b/c have the same remainder. For example, 8 = 2 (mod 3), because 8/3 and 2/3 have remainder 2.
One property of this relation is that for any integer x and for any nonzero integer y, there exists a unique integer z such that x = z (mod y) and z is between 0 and y inclusive. The upshot of this is that, in most cases, if you know how your relation behaves with all integers between 0 and y, you know how it behaves for all integers.
Consider the quadratic residues mod 8; that is, find all possible values of c if 0 <= c < 8 and x2 = c (mod 8) for some integer x. Plugging in all values from 0 to 7 for x, the only possible values of c are 0, 1, and 4.
Now consider 2n + 1 (mod 8). We know that 2n + 1 is a perfect square, so we know that 2n + 1 = 0, 1, or 4 (mod 8). Thus, 2n = -1, 0, or 3 (mod 8). Since 2n is an even number, and 8 is an even number, 2n can only be congruent to an even number mod 8. Therefore, 2n = 0 (mod 8), and therefore n = 0 (mod 4).
Finally, consider 3n + 1 (mod 8). As before, we note that 3n = -1, 0, or 3 (mod 8). We know that n = 0 (mod 4), so we know that n = 4k for some integer k. Therefore, n is even. Since 8 is also even, we know that n = 0 (mod 8). Therefore, n is divisible by 8. QED.
Either diagonal of a parallelogram divides the parallelogram into two triangles of equal areas. Thus area of triangle abd = half that of the parallelogram abcd. The required ratio is 1 / 2.
About how much did one square inch of the drawing paper weigh?
I'm not totally sure what's being asked here, but I'll take a shot at something and see if I hit anything. What we know: * An 8 x 10 piece of paper weighs 9.8 grams. * A cut-out of a leaf is 20 square inches. The 8 x 10 paper has an area of 8 x 10 = 80 square inches. The cut-out is only 20 square inches. What fractional part of the whole paper is the cut-out? Just divide 20 by 80: 20/80 = 1/4 = 0.25 So, the area of the cut-out is about one quarter the area of the whole paper, so we can calculate the weight of the cut-out by multiplying 0.25 by 9.8 grams: 0.25 x 9.8 = 2.45 grams
If A is an orthogonal matrix then why is it's inverse also orthogonal?
First let's be clear on the definitions.
A matrix M is orthogonal if MT=M-1
Or multiply both sides by M and you have
1) M MT=I
or
2) MTM=I
Where I is the identity matrix.
So our definition tells us a matrix is orthogonal if its transpose equals its inverse or if the product ( left or right) of the the matrix and its transpose is the identity.
Now we want to show why the inverse of an orthogonal matrix is also orthogonal.
Let A be orthogonal. We are assuming it is square since it has an inverse.
Now we want to show that A-1 is orthogonal.
We need to show that the inverse is equal to the transpose.
Since A is orthogonal, A=AT
Let's multiply both sides by A-1
A-1 A= A-1 AT
Or A-1 AT =I
Compare this to the definition above in 1) (M MT=I)
do you see how A-1 now fits the definition of orthogonal?
Or course we could have multiplied on the left and then we would have arrived at 2) above.
Is this valid all x is y some y is z some x is z?
No. The way this is worded all x are y but not all y are necessarily x.
.
Example:
All Gorillas are Apes.
Some Apes are Chimpanzees.
Some Gorillas are Chimpanzees. (Not True)
.
All Dogs are Mammals.
Some Mammals are Cats.
Some Dogs are Cats. (Not True)
.
Analysis:
All A are B.
Some B are C.
Therefore, Some A are C.
.
∀x(Ax → Bx)
∃x(Bx ∧ Cx)
∴ ∃x(Ax ∧ Cx)
Truth Tree
..............................|Bn ET(n)
..............................|Cn ET(n) premise
............................./.\
.........................../....\
.......UT prem ~An / ......\
........................../\........\
......................../..\.........\
UT neg con ~An /.....\ ~Cn \ Bn UT prem
.............invalid ↑..... ↑...... /\
...................................../...\
.............................~An /......\ ~Cn UT negative conclusion
...................................↑.......x Invalid
Ignore the periods. The spaces were being deleted so I included periods to make the truth tree readable.
What is the water contain in human blood?
Water is water, doesn't matter where it comes from. The fluid that has water and lots of other stuff in the blood is plasma.
What is the density of pure water?
Study shows that pure water density is 1gm/cc. This might be at lower altitudes because that 1gm could not be 1gm anymore at high altitudes. Salt water density is slightly higher thus it was said that ships float better at sea meaning that sea water has more bouyant force than pure water. 8^D
It depends on the temperature. Water is densest at 4 deg. Celsius. It is actually slightly less dense if you get colder or warmer.
But at 4 deg. C. the density is 1.000 g/mL
Reasons why pyramids and prisms have nothing in common?
They are both polyhedra. Therefore the question is based on false premises and so is a waste of time.
What theorems are expected to come up in the junior cert higher level maths 2010?
Pythagoras is supposed to come up..! and also my teacher said it could be the first theorem! but don't forget there is constructions also and I'm not completely sure about those Thermos's. I hope this helped..!
What are the irregular quadrilaterals?
Rhombuses, rectangles, kites, parallelograms, and trapezoids are all irregular quadrilaterals.
How many times can a paper money be folded before it tears?
There is no single answer. Different countries use papers of different quality - or even plastics.
Why square root of negative 36 is not a real number?
When you multiply any number by itself, the result is always positive. You cannot take any real number and square it to get something negative. The square root of -36 is 6i.
