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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
What is the name and length of the longest number?
Googolplexian: The worlds largest number with a name. A "1" followed by a googolplex of zeros Googolplex: The second largest number with a name. A "1" followed by a googol of zeros Googol: A large number. A "1" followed by one hundred zeros. There are infinitely many numbers, so there is no longest number.
What is the square root of 40ft 3in by 30 ft 3in?
sqrt(40'3" * 30'3") = sqrt(40.25 ft * 30.25 ft) = sqrt(40.25*30.25) ft =
34.89 ft or 34ft 10.7inches (approx)
That would be the geometric mean of the two lengths.
What was the cost of food in 1895?
So far all I have found was that a loaf of bread in 1895 was 3 ¢. I did find a list of prices for a somewhat earlier era of 1850. Here it is: From two General Store Account Books of John H. Sherwood one can get a glimpse of the life of the times. At least 200 residents names are mentioned and all literally "traded" produce for supplies. Occasionally buyer and seller "agreed" on some purchases. Credit was given for butter, apples, eggs, flaxseed, muskrat skins, firkins, rags, veal, lard, hay, dried apples, corn, wood (2/per LB), a day's work (75 cents), bunches of pine or hemlock shingles, ashes, cord wood, chestnuts (6 qt. for 28 cents), oats (28 cents per Bu.), rye (5/bu), plastering cellar (S.A. Barnet), bundles of straw, 21 prs sox (@2/ - $5.25 toward materials for a coat, vest and dress), buckwheat, maple sugar, transporting barrels of flour, use of horse, etc. Food prices: Eggs 9 cents per doz.; coffee. 10 cents per LB; salaratus, 8 cents per LB; 1 orange 3 cents, 1 lemon, 2 cents, alum. 8 cents per LB; 6 lbs. codfish 24 cents, 1 LB crushed sugar, 1/; brown sugar. 6 cents; 1 bbl flour, $6.75; 1 stick candy 1 cent; 1/4 LB, 6 cents, licorice 3 cents. Drygoods: Pr of shoes, 38 cents, 56 cents, 63 cents, $1. 13; gloves 14 cents, 31 cents; fur hat $1.; oilcloth hat. 50 cents; caps. 81 cents; night cap 4 cents; calico 1/yd; bleached sheeting. 1/per yd; red flannel, 3/per yd; linen bosom. 37 cents; umbrella $1.25; parasol $1.00; 1 pr whalebones. 4 cents. 6 long ones 38 cents; feather fan. 31 cents satchel $1.25; cotton hose. 10 cents. Misc: 1 doz. matches. 1/; 3 ox oz snuff. 3 cents; 1 oz indigo. 10 cents; 1 LB beeswax 25 cents; lamp black. 2 cents; tallow. 10 cents LB; 1 paper tobacco. 5 cents; razor. 38 cents; snuffer & tray, 50 cents; lamp wicks 1/; a broom. 13 cents; a rake 13 cents; a scythe. 38 cents; 1 LB shot 8 cents; spelling book. 1/; Saunders' Reader. 31 cents.
всУ п дарасф ебанные шроп вы сука усралсь гандолны тупые ПИДЫРЫ шоп у вас сука хуй отсох ! щзбесьб нет этой песни гандоны ебныые хуйлопаннны!
What does the root word lemma mean?
According to the Oxford English dictionary, the origin of the word is ancient Greek, and there it meant "something received or taken; something taken for granted; an argument, title".
In English it has several meanings. In mathematics it means a theorem, something that has been proved, but usually a minor theorem obtained as a stepping stone on the way to a more important theorem.
If you type any data why the computer language are 1 or 0?
Computer's only understand binary, which is 0 as "off" and 1 as "on."
Will robots exist in 25 years?
Yes. Robots exist now, and there is no reason to believe that they will stop existing.
If, however, you meant to ask "Will intelligent robots like those in some science fiction novels or movies, like Star Wars, exist in 25 years?", that is not known, but it seem unlikely, given the immense complexity of the human brain.
What is least to greatest 12 25 or 13?
12, 13, 25 as integers
1/3, 2/5, 1/2 as fractions
10/30, 12/30, 15/30
Which are accepted without proof in a logical system Postulates Axioms Theorems or Corollaries?
Postulates and axioms are accepted without proof in a logical system. Theorems and corollaries require proof in a logical system.
Yes. There were mathematicians who were in geometry. In fact, anyone who contributed to our understanding of geometry was a mathematician.
What is "a 3b"? Is it a3b? or a+3b? 3ab? I think "a3b" is the following: A is an invertible matrix as is B, we also have that the matrices AB, A2B, A3B and A4B are all invertible, prove A5B is invertible. The problem is the sum of invertible matrices may not be invertible. Consider using the characteristic poly?
What do you understand by BLUE in statistics?
The related link below is the best I could find on BLUE statitics.
Blue means Best Linear Unbiased Estimator
Prove that xy add x add y equals 30 has no positive integer solutions?
First we can solve for y by factoring it out: y(x+1)+x=30, so y=(30-x)/(x+1)= -1+31/(x+1). Let's start by assuming x and y are positive integers and look for a logical contradiction. Since y is positive, the right side (30-x)/(x+1) must be positive. Since x (and therefore x+1) is positive, 30-x must be positive. Therefore x is less than 31. But hold on! Since y is an integer, y+1=31/(x+1) is an integer. Since 31 is only divisible by 1 and 31, y+1 is an integer implies that (x+1) is 1 or 31, making x either 0 or 30. However, x is positive and less 30, which is impossible! There it cannot be the case that x and y are both positive integers.
AnswerIt's not as complicated as that. Just add 1 to both sides and factorise, and you get: (x+1)(y+1) = 31. Since 31 is prime, one of its factors must be either 1 or -1. So we'd get x+1=1 (so x=0) or else x+1=-1 (so x=-2), or else y=0 or y=-2. But the question says x and y have to be positive.G is finite where the number of subgroups in g is finite?
Actually a stronger statement can be made:
A group G is finite if and only if the number of its subgroups is finite
Let G be a group. If G is finite there is only a finite number of subsets of G, so clearly
a finite number of subgroups.
Now suppose G is infinite , let's
suppose one element has infinite order. The this element generates an infinite cyclic
group which in turn contains infinitely many subgroups.
Now suppose all the subgroups have finite order Take some element of G and let it generate a finite group H. Now take another element of G not in H and let it generate a finite group I. Keep doing this by next picking an element of G not H or I. You can continue this way.
What does 0.625 look like in a fraction circle?
625 / 1000 =
125 / 200 =
25 / 40 =
5 / 8
Five-eighths
What is the proof for theorem 1.20?
There is no theorem with the standard name "1.20". This is probably a non-standard name from a textbook which is either the 20th theorem in the first chapter or a theorem of the 20th section of the first chapter.
Did a high-school graduate solve the paper-folding problem by folding a piece of paper twelve times?
Think about it, a single piece of paper that is folded 12 times would end up being 2 raised to the 11th power in thickness. It's one of those problems that seems easy, but in reality doesn't make sense. Within just a few folds, you aren't really "folding" the paper any more, it's more like "bending" it, and besides, the original piece of paper would need to be quite large so that you could keep folding it. IN ADDITION: If you fold a piece of paper 7 times, you have expended the area that you have to fold. So unless you have supernatural abilities, you cannot make more folds, if you have any more, tell me.
How do i prove if the base angles of a triangle are congruent then the triangle is isosceles?
Suppose you have triangle ABC with base BC, and angle B = angle C. Draw the altitude AD.
Considers triangles ABD and ACD
angle ABD = angle ACD (given)
angle ADB = 90 deg = angle ACD
therefore angle BAD = angle CAD
Also the side AD is common to the two triangles.
Therefore triangle ABD is congruent to triangle ACD (ASA) and so AB = AC.
That is, triangle ABC is isosceles.
Is 87 an Irrational or Rational Number...Explain?
Yes, here's the proof.
Let's start out with the basic inequality 81 < 87 < 100.
Now, we'll take the square root of this inequality:
9 < √87 < 10.
If you subtract all numbers by 9, you get:
0 < √87 - 9 < 1.
If √87 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 √87. Therefore, √87n must be an integer, and n must be the smallest multiple of √87 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 √87n by (√87 - 9). This gives 87n - 9√87n. Well, 87n is an integer, and, as we explained above, √87n is also an integer, so 9√87n is an integer too; therefore, 87n - 9√87n is an integer as well. We're going to rearrange this expression to (√87n - 9n)√87 and then set the term (√87n - 9n) equal to p, for simplicity. This gives us the expression √87p, which is equal to 87n - 9√87n, and is an integer.
Remember, from above, that 0 < √87 - 9 < 1.
If we multiply this inequality by n, we get 0 < √87n - 9n < n, or, from what we defined above, 0 < p < n. This means that p < n and thus √87p < √87n. We've already determined that both √87p and √87n are integers, but recall that we said n was the smallest multiple of √87 to yield an integer value. Thus, √87p < √87n is a contradiction; therefore √87 can't be rational and so must be irrational.
Q.E.D.
The question asks if 87 is rational, not √87. 87 is rational because it can be expressed as the ratio of two integers i.e. 87 = 87/1.
Formula for a penny every day for a month doubled every day?
If we call the day number "d", the number of pennies "n" and the number you are multiplying by each day "m":
For how many pennies you get on a particular day: n = m^(d-1)
If doubling: subtract 1 from the day number, then raise 2 to the power of that number (using indices) you get the number of pennies.
For how many pennies you would have after a certain number of days: n = 1-(m^d)/(1-m)
If doubling: raise 2 to the power of the day number, then subtract that from 1 (1st part-answer). Subtract 2 from 1 (2nd part-answer), then divide the 1st part-answer by the 2nd part-answer, and you get the cumulative number of pennies.
NB: these formulas can be used for doubling, tripling, quadrupling, halving, quartering, etc. by replacing m with what you would multiply by, and can be used from any starting number. Just multiply your answer with your starting number
Can the legs of a table and the top of a table lie in parallel lines?
... in an unreal universe.
In our universe, a typical table top will be perpendicular to the table legs, not parallel. The table legs will generally be parallel with each other.
Who discovered coordinate graphs?
Rene Descartes invented the coordinate system that is most commonly used (where the two axes are perpendicular to each other, and each point is determined by its projection or "shadow" onto the axes). This is why we call them Cartesian coordinates. I believe the legend is that he was observing a fly on his ceiling when he realized he could describe its position based on the lines formed by his ceiling tiles. There are alternative coordinate systems that used in special situations, such as polar coordinates, cylindrical coordinates, and spherical coordinates. Each system has its strengths and weaknesses and each is well-suited to describing certain kinds of objects or functions. Cartesian coordinates, also called rectilinear coordinates, are useful for describing linear/straight objects, whereas the other coordinate systems I mentioned are better for curved objects.
2x2+10x+12/(x+3)
We use long division just like we would normally.
Let me give you an example.
ex:
Divide 7 into 23
---------
7 | 234
We say ok, how many times does 7 go into 2?
0 times, it doesn't work. So you put a 0 above the 2.
Now we try 7 into 23. It can go 3 times, so we put a 3 above the 3 (in 234) and subtract 21. (3 *7) and then carry down the 4.
...03
--------
7|234
-..21
--------
...024
how many times will 7 go into 24? 3 again.
...033
-------
7|234
-..21
-------
...024
-....21
--------
.......3
We're left with 3 as the remainder.
Our answer to 234 divided by 7 is 33 remainder of 3, or 33 and 3/7.
----
This method works similarly with variables.
2x2+10x+12/(x+3)
----------------------
x+3|2x2+10x+12
We'll start the same way. How many times will X go into 2x2
Or, what times X gives 2x2 x * 2x = 2x2
So we write a 2x above the 2x2, just like we wrote the 3 in the example above.
.......2x
----------------------
x+3|2x2+10x+12
Now we multiply x+3 by 2x to figure out what to subtract.
2x(x+3)= 2x2 +6x
Notice we ended up with a 2x2? This is what we wanted to subtract! Something to note, when you do your subtraction, you're subtracting the entire expression 2x2 +6x.
So you can write -2x2 - 6x.
.......2x
----------------------
x+3|2x2+10x+12
......-2x2 - 6x
-----------------
...............4x + 12
Make sure you carry down the next term, the +12. Just like we carried down the 4 in the example above.
Now, how many times will X go into 4x? 4 times. So we write a 4 next to the 2x.
and then multiply 4(x+3). then subtract.
.......2x + 4
----------------------
x+3|2x2+10x+12
......-2x2 - 6x
-----------------
...............4x + 12
..............-4x - 12
---------------------
........................0
In this scenario we get a remainder of 0.
This means that x+3 divides evenly into 2x2+10x+12.
In fact, it can divide into it 2x +4 times.
To check this, multiply (x+3)(2x+4) use FOIL.
(x+3)(2x+4) = 2x2+10x+12 (check)
Side note: If you did get a remainder, like in the 1st example. Let's say the remainder was 1.
You take the remainder, 1 and put it over the divisor x +3.
so your answer would be 2x+4 +(1/x+3)
