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You are most welcome.
Now you will like this: There are 10 kinds of people - those who know binary and those who don't.
In binary, the values of the first four positions, from right to left, are 1, 2, 4, 8.
So 1101 (base 2) means you have, from right to left, one 1, no 2's, one 4, and one 8. Add them up and you have the value of 1101 (base 2).
Do the same for the other number, and then (assuming the ^ indicates exponent) apply the rule for raising to a negative exponent, which I presume you can find if you don't remember it.
I assume by "solve" you mean "simplify," as it is not an equation.
I find two things confusing in your expression:
1. Is the small "x" before the radical a "multiply" ? If so, you can delete it. Pretty sloppy notation.
2. Is the "1/2" under the radical supposed to be the X's exponent?
There is one more interesting category.
A number is called algebraic if it is the root of a polynomial equation with rational coefficients.
Non-algebraic numbers are called transcendental.
So sqrt(5) is (as you correctly say) irrational, but it is algebraic, being the root of x^2 - 5 = 0.
More complicated to prove: pi and e are not just irrational, but also transcendental.
Roster method is a common term for just listing the elements.
While we're talking, note that order does not matter; an element is either in or not in a set.
So {a,b,c} = {b,a,c} and any other permutaion.
Bob adds another condition for axioms I should have mentioned - that of independence from the others.
It is considered elegant for the set of axioms for a theory to be as small as possible.
For centuries mathematicians tried to prove Euclid's parallel axiom from the others. It somehow seemed less elementary than the others. Some thought they had, but they had erred.
Starting in the 19th century, the creation of non-Euclidean geometries (i.e., models of geometry with other forms of the parallel axiom, yet with most or all of the others) proved (from outside of the axiom system) that the parallel axiom was in fact independent of the others.
When mathematicians develop an axiom system for a subject, e.g., the axioms for plane geometry, or the axioms for set theory, two main criteria are:
1. the axioms must not contradict each other
2. the axioms must encompass what is considered to be the body of knowledge of the subject
Especially number 2 - this sort of seems like hand waving. But a most interesting example is the Zermelo-Fraenkel plus Axiom of Choice axioms for set theory (ZFC), developed and agreed to over years as encompassing what is considered to be a full description of set theory.
In 1900 Hilbert's first problem posed for the upcoming century is known as the continuum problem. In short, the problem is: is the size of the set of real numbers the very next transfinite size (i.e., aleph-1) after the size of the natural numbers (aleph-0).
BUT - In 1963 Paul Cohen proved - wait for it - you can have it either way. Either statement (Reals is aleph-1 or Reals is greater than aleph-1) is consistent with ZFC. So either statement is independent of ZFC.
You can find all of this in Wikipedia or any book on set theory.
Depends on whether the sign means anything in the context of what the numbers represent.
E.g., if I owe you $10, then my balance with you is -10. Now I repay you $2, so +2 - 10 = -8, my new balance with you.
Other examples might be like comparing how far the two of us walk. Do we care about just the absolute value, or care whether you go North (+) and I go South (-) ?
All hexagon edges are straight lines, rather than the smoothly changing circumference of a circle, so the formula is based on summing various triangles than make the hexagon.
The formula for the area of a circle is derived from the integral calculus, which deals with smoothly changing shapes.
Hard to explain otherwise.
A terrible idea. Visualizing slope is so important, especially to beginners.
My hand-held graphing calculator also has non-equal vertical and horizontal spacing, which I point out to tutees and urge them to use websites that do this correctly.
All possible outcomes:
First flip is either H or T - 2 events.
For each of these 2 events, second flip is either H or T - 2 events. So far, for 2 flips: HH, HT, TH, TT; count = 2*2.
For each of these 4 events, ditto; count = 4*2 = 2*2*2 = 2^3.
How did your app solve the problem and what were the results?
Sometimes the results might be presented differently, yet be equal.
The result of each flip is independent of the others. Each flip has 2 equally likely outcomes - assuming a fair coin.
So 2^3 = 2x2x2 = 8 equally likely outcomes.
You might also say that 10,24,26 is scaled up from 5,12,13, another well-known right triangle, but certainly not as cool as 3,4,5.
Conventions only work if everyone agrees. Rounding up has been the convention. For most situations, everyone agrees.
That having been said, there are special cases when many would consider, or the actual rules say, that rounding up wrong.
If a baseball player's average is .3995, that may round to the coveted .400, but I don't think there are bragging rights here.
In bowling, from the internet (!), "If the result is a decimal, you typically round down to get to a whole number."
So, what did you get and what does the book say?
What do you need to multiple the denominator by so that it is an integer? Multiple the numerator by the same.
For infinity discussion, I refer you to my post: https://www.mathisfunforum.com/viewtopic.php?id=31062
I agree - long division. Synthetic, while a cool idea that it works, is unnecessary, and for students with so much else to learn in the course, just a lot of clutter.
I'll add a bit more to Bob's excellent exposition.
The expression under the radical (the radicand) is called the discriminant.
Note that even before completely solving for the roots, the value (<0, =0, >0) of the discriminant tells us something about the roots and the graph.
=0: both roots are the same, because the formula has ±0. The parabola's vertex touches the x-axis there.
<0: both roots are complex; they are called complex conjugates: a+bi and a-bi. (Any two expressions x+y and x-y are called conjugates.) The parabola does not intersect the x-axis.
>0: both roots are real. The parabola has two distinct x-intersects.
You say the circle is shaded, and that you want the area of the shaded part.
So - you want the area of the circle, not the difference between the areas of the square and the circle.
Also, technically, you can calculate this only if the circle is specifically inscribed in the square, not just inside it. An infinite number of circles of various radii are inside the square.
I'm not being picky here, just using mathematical terminology correctly.
Well, I do apologize. I was expecting a thanks for pointing forum members to what I consider a most interesting field.
In short: two sets are considered to have the same size (or cardinality) if they can be put into a 1-1 relationship.
E.g., the naturals N = {1,2,3,...} would seem naively to be twice as big as the evens E = {2,4,6,...}. However, as they can be matched up 1-1: (1,2), (2,4), (3,6), and so on, they have the same size. Georg Cantor called this smallest transfinite size by the first Hebrew letter aleph, with a subscript zero, read as aleph-null.
It turns out that many infinite sets are in fact aleph-null sets.
Take the rationals Q = {p/q where p and q are integers}. Although there are an infinite number of rationals between any two integers, Q is in fact aleph-null. Here's a listing that contains all rationals (some may be duplicated): {1/1, -1/1, 1/2, -1/2, 2/1, -2/1, 2/2, -2/2, 1/3, -1/3, 2/3, -2/3, etc.}. You should get the point here. Since they can all be listed, they match 1-1 with N, hence aleph-null (also called countably infinite).
BUT - the reals are > aleph-null.
Proof by reductio ad absurdum. Assume you have a list of all reals. Then here's a real not in your list: Construct it by creating a number that differs from the nth number in your list in its nth position. Since this new number is not in your list, you did not in fact provide such a list, hence it cannot be done. Even if you add that number to the list, we'll just do this all over again.
Here's the really cool part: In 1900 or so David Hilbert created his famous list of 23 problems for the next century. First on the list: the Continuum Problem: is the cardinality of the reals aleph-one, i.e., the very next transfinite size?
In 1963 Paul Cohen proved (get ready for this) - You can have it either way. That is, the statement "size of the reals is aleph-one" and its negation are independent of the generally accepted axioms of set theory, ZFC (Zermelo-Fraenkel with Axiom of Choice).
I see numerous (and in my opinion repetitive and meaningless) posts about doing various arithmetic with infinity: ∞.
The theory of transfinite arithmetic was initiated mainly by Georg Cantor around the 1890's.
The theory explores in a definitive and axiomatic way the various (differently sized) infinite sets of different types of numbers (e.g., Natural Numbers, Rationals, Irrationals, Transcendentals).
This is too involved to say more here, but a good starting point are these articles:
https://en.wikipedia.org/wiki/Aleph_number
https://en.wikipedia.org/wiki/Transfinite_number
https://en.wikipedia.org/wiki/Continuum_hypothesis
But 2.151515 is not in your set N.