Proofs

acute angle - area

chord - cone - circumference - circle - cube - cylinder - closed curve - congruent figures - compass

edge - equilateral triangle - ellipse

helix - hexagon - hypotenuse

intersecting lines - isosceles triangle

parallel lines - parallelogram - plane - polygon - pentagon - pyramid - point - perimeter - Pythagorean theorem - protractor

ray - rhombus - rectangle - right angle - right triangle - radius -

square - scalene triangle - sphere - symmetry line

vertex - volume

trapezoid - triangle

Yes. Simple example:

a=(1 i)

(-i 1)

The eigenvalues of the Hermitean matrix a are 0 and 2 and the corresponding eigenvectors are (i -1) and (i 1).

A Hermitean matrix always has real eigenvalues, but it can have complex eigenvectors.

Yes, but only if any language in NP has small uniform circuits.

Hope this helps. This is a famous unsolved problem, as whoever posted it here probably knows. A prize of one million US dollars has been offered for a solution. There are details at http://www.claymath.org/millennium/P_vs_NP/ At the top right of this page there is a link to the "Official problem description" by Stephen Cook, one of the people who first posed the problem in 1971, and also a more informal description under the heading "Minesweeper". Anyone who solves this problem will get not only a million dollars, but also enduring fame among mathematicians and computer scientists. It won't be easy, since a lot of clever people have worked on it since 1971. Most people believe that P is not equal to NP, but a belief is not the same as a proof.

There are 5 groups of order 8 up to isomorphism. 3 abelian ones (C8, C4xC2, C2xC2xC2) and 2 non-abelian ones (dihedral group D8 and quaternion group Q)

rutrags defreds

newton and Leibniz were first introduced the concept of limit independently

z varies directly as the square of r

So z = cr2 where c is the constant of variation.

z = 24 when r = 4 ie 24 = c*42 so that c = 24/42 = 1.5

When z = 121.5,

121.5 = 1.5*r2 so that r2 = 121.5/1.5 = 81 so that r = -9 or +9

Yes; it will form a honey-comb shape.

<http://gwydir.demon.co.uk/jo/tess/bighex.gif> for image.

Not if you want REAL solutions:

y=4x^4 +2x^2 +3

t=x^2

therefore y=4t^2 +2t +3

Use discriminant: b^2-4ac=4-48=-44

this means there will be no REAL solutions (but there will be COMPLEX solutions-message me if you need this bit explained)

It depends what you mean by "necessary". There is a choice of different systems for (classical) predicate logic, but they all give the same results.

Universal introduction is certainly a valid principle in predicate logic, so the question is: Does universal introduction have to be one of the basic rules of the system?

The answer is no. It can be a derived principle. It is even possible to introduce "for all" as a derived symbol, and only have "there exists" in the basic system. The basic system would have a couple of rules controlling "there exists", and from these rules universal introduction would be a derived principle.

There is no solution. The answer to this sort of question (such as y=2x+3, and y=2x+4) would be no solution, since the lines never intersect, but instead continue on to go an infinite distance without ever crossing each other.

The perfect square roots are simply the counting numbers: 1, 2, 3, 4, and so on.
The square root of 1 is 1, the square root of 4 is 2, the square root of 9 is 3, the square root of 16 is 4, and so on....
See http://www.naturalnumbers.org/psquares.html

An infinite number of planes. Picture a line and now picture planes going in every direction through the line,

16, in four sets of four. Plus expiry dates, and security numbers at the back.

Yes; all parallelograms have diagonals that bisect each other.

Other properties of parallelograms are:

* The opposite sides are congruent. * The opposite sides are parallel. * The opposite angles are congruent.

Cayley's Theorem states that every group G is isomorphic to a subgroup of the symmetric group on G.

The digit 9 in the number 89123 represents nine thousands.

The circumcenter of a triangle is equidistant from the vertices of a triangle.

Prove it by induction on n, use 0 or 1 as base cases.

cuz a resistor is basically a long wire , that resists some of the charges flowing across it.

it has resistive properties which stops some of the current from flowing from one terminal to the other.

so....as it is just a piece of wire....it doesn't need a +ve or a -ve terminal

quite big, maybe