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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
The binomial theorem describes the algebraic expansion of powers of a binomial, hence it is referred to as binomial expansion.
Is it unethical to purposely distort communication to get favourable outcome?
The answer depends on the situation.
In baseball, when the catcher signals the pitcher with finger signals, he is concealing his communication from the other team.
That is not unethical.
Without such justification, the answer seems to be yes.
How does halving a sample affect the weight of the sample?
If the sample is homogeneous, then half of its volume has
half of its mass and half of its weight.
Why are the diagonals of a parallelogram not congruent?
If you look ate the parallelogram you'll see two kinds of triangles. Two that have longer diagonal and bigger angle, and two sides of parallelogram. Then, you have two triangles that have two sides of parallelogram, shorter diagonal and smaller angle. This triangles obviously have two sides that are the same (sides of parallelogram). If this two triangles had been congruent diagonals would have been congruent too, since these triangles would have been congruent. But this is not true unless angles of parallelogram are the same, therefore diagonals cannot be the same length. Of course, there are parallelograms that have same angles, and those are square and rectangle, which do have the same angles. I hope I made this more clear, and I'm sorry for my bad English.
Do the two diagonal bisect each other?
2 diagonals bisect each other only in the case of square , parallelogram, rhombus , rectangle and isosceles trapezium ;not in ordinary quadrilaterals.
Can the Third angle theorem prove that the ASS triangle is congruent?
No, because the third-angle theorem requires that you know two angles of each of the triangles. Assuming ASS, you only know one angle.
In fact, two triangles may have the same ASS and not be congruent. See if you can make two non-congruent triangles with Angle 60 deg, Side 10, and Side 9.
What do you call a statement in geometry that requires proof?
Theorems are statements in geometry that require proof.
An algebraic way of turning a mixed number into a fraction is (as described in your question): a b/c = (ac + b)/c
This works because ac/c = a (the c will cancel itself out.) just like how 50/50 = 1, and (5*15)/5 = 15. You're basically multiplying a by c/c, which is 1.
When you add fractions with the same denominator, you simply add the numerators together.
Thus, a b/c = ac/c + b/c = (ac + b)/c
What is the proof of the ''Fundamental Theorem of Algebra''?
The Fundamental Theorem of Algebra:
If P(z) = Σnk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0, then P(z0) = 0 for some z0 Є C. Descriptively, this says that any nonconstant polynomial over the complex number space, C, can be written as a product of linear factors.
Proof:
First off, we need to apply the Heine-Borel theorem to C. The Heine-Borel theorem states that if S is a closed and bounded set in an m-dimensional Euclidean space (written as Rm), then S is compact.
From above, P(z) = Σnk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0. Let m = inf{|P(z)| : z Є C} where inf is the infinum, or the greatest lower bound of the set.
From the triangle inequality, |P(reit)| ≥ rn(|an| - r-1|an-1| - … - r-n|a0|),
so limr --> ∞ |P(reit)| = ∞. Therefore there is a real number R that |P(reit)| > m + 1 whenever r > R.
If S = {reit : r ≤ R}, then S is compact in C, by the Heine-Borel Theorem; and let m = inf{|P(z)| : z Є S}. |P| is a continuous and real-valued function in S, so, using the result from another proof not done here, it has a minimum value on S; i.e., there is a value for z0 Є S that makes |P(z0)| = m. So, if m = 0 then the theorem is proved.
We're going to show that m = 0 by proving that m can't equal anything else, and since we know m exists, it has no choice but to be zero. So, suppose m ≠ 0 and let Q(z) = P(z + z0)/P(z0), z Є C.
Q is therefore a polynomial with degree n and |Q(z)| ≥ 1 for all z Є C.
Q(0) = 1 so Q(z) can be expressed via P's series as:
Q(z) = 1 + bkzk + … + bnzn where k is the smallest positive integer ≤ n such that bk ≠ 0.
Since |-|bk|/bk| = 1, there exists a t0 Є [0, 2π/k] such that eikt0 = -|bk|/bk.
Then Q(reit0) = 1 + bkrkeikt0 + bk+1rk+1ei(k+1)t0 + … + bnrneint0
= 1 - rk|bk| + bk+1rk+1ei(k+1)t0 + … + bnrneint0.
So, if rk|bk| < 1 then |Q(reit0)| ≤ 1 - rk(|bk| - r|bk+1| - … - rn-k|bn|).
That means that if we pick a small enough r, we can make |Q(reit0)| ≤ 1 which contradicts the statement above that |Q(z)| ≥ 1 for all z Є C. Therefore m ≠ 0 doesn't hold and P(z0) = 0
Q.E.D.
Another proofSuppose P has no zeroes. Then we can define the function f(z) = 1 / P(z), and f is analytic. By the proof above, P(z) tends to infinity as z tends to infinity; hence f(z) tends to 0 as z tends to infinity. So there is a disc S such that f, restricted to the outside of S, is bounded. Also by the proof above, f is bounded inside the disc as well; therefore f is bounded. Now we apply a theorem called Liouville's Theorem, which says that any analytic function which is defined on all of C and is bounded must be a constant. So f is a constant; therefore P is constant. But we were assuming that P is not constant, so this is a contradiction.(To prove Liouville's Theorem: Suppose M is a bound for the function f, i.e. |f(z)| < M for all z. Suppose a and b are complex numbers, and we want to show f(a) = f(b). Use the theorem that f(a) = integral of f(z)/(z-a) / (2 * pi * i) around the circle of radius R and centre 0. Then, if R is sufficiently large:
|f(b) - f(a)|
= | integral, around circle, of (f(z) * (1/(z-b) - 1/(z-a))) | / (2*pi)
= | integral around circle of (f(z) * (b-a) / ((z-a)(z-b)) ) | / (2*pi)
<= (M * |b-a| / ((R-|a|)(R-|b|)) ) * (2*pi*R) / (2*pi)
The last line uses the formula |integral| <= |pathlength| * |maximum value|. Then we get |f(b) - f(a)| <= M * |b-a| * R / ((R-|a|)(R-|b|)). Letting R tend to infinity, we can prove that |f(b)-f(a)| is as small as we like; therefore f(a) = f(b).
)
How many 10-digit even numbers can be formed if the digits can be repeated?
Assuming that leading zeros are not permitted (that is the lowest ten digit number is 1 000 000 000), then there are 4 500 000 000 possible even numbers
Applications of differentiation in real life?
1. It is used ECONOMIC alot,calculas is also a base of economics
2.it is used in history,for predicting the life of a stone
3.it is used in geography ,which is used to study the gases present in the atmosphere
4. It is mainly used in daily by pilots to measure the pressure n the air
How do you prove if the given function is a constant function using Mean Value Theorem?
The Mean Value Theorem states that the function must be continuous and differentiable over the whole x-interval and there must be a point in the derivative where you plug in a number and get 0 out.(f'(c)=0). If a function is constant then the derivative of that function is 0 => any number you put in, you will get 0 out. Thus, using the MVT we deduced that the slope must be zero and since the f(x) is a constant function then the slope IS 0.
No, only natural numbers (positive integers) can be prime.
Why is the troposphere called the turbulent sphere?
The troposphere is also called the turbulent sphere because it is the sphere with the most change. It has moving air currents, clouds, storms, jet streams, strong and other weather phenomena that affect weather patterns.
Can theorems be used to prove other theorems?
Yes, they can. This is done all the time in mathematics, logic and other areas. However, you must ensure that you either record the theorems used, or write them out in whole and attach them to the proof of the new theorem.
Who proved pi to be irrational?
The value of pi has never been proven becauase it is an irrational number which can not be expressed as a fraction
What is Proof of circle theorem in fluid?
Since z = a^2/z on the circle, we see that w as given by (1) is purely
real on the circle C and therefore if* = 0. Thus C is a streamline.
If the point z is outside 0, the point az
/z is inside 0, and vice-versa. Since
all the singularities off(z) are by hypothesis exterior to C, all the singularities
off(a?/z) are interior to C ; in particular f(a
z
/z) has no singularity at infinity,
since f(z) has none at z = 0. Thus w has exactly the same singularities as/(z)
and so all the conditions are satisfied.
Explain why you can not use angle angle angle to prove two triangles are congruent?
Knowing that three angles are congruent only proves that two triangles are similar. Consider, for example, two equilateral triangles, one with sides of length 5 and the other of lengths ten. Both have three angles of 60 degrees each, but they are not congruent because their sides are not of the same length.
Can Anyone Prove 100 minus 100 divided by 100 minus 100 equals 2?
(100 - 100 / 100 - 100) = 2
=> 102 - 102 / 10(10 - 10) =2
=> (10 + 10) (10 - 10) / 10(10 - 10) = 2 [a2 - b2 = (a + b) (a - b)]
Cancelling ( 10 - 10 ) {numerator} from ( 10 - 10 ) {denominator}
=> ( 10 + 10 ) / 10 = 2
=> 20 / 10 = 2
=> 2 = 2 [Hence, Proved]
False.
The angles can be formed by two skew lines intersecting a third line.
Which cannot be used as a reason in a proof?
AAA (angle angle angle) cannot be used as a reason in a proof when proving triangles congruent .
208 times,
from 1 to 100 there r 11 nine such as 9 , 19 , 29, 39, 49,59, 69,79, 89, 99 its prove there are 11 nines so up to 800 there are 11*8=88 nines and from 900 to 1000 there are 100(901, 902,903......)+ 10 ( 990,991,992.....)+10(909,919,929,939....)=120 that means there are in total 88+120=208 nine in 1 to 1000
