You are not logged in.
As a tutor, I also point out to students that math rules typically do not depend on the form in which a number is represented - i.e., the numerals.
So we are correctly taught that to divide by a fraction, we multiply by the reciprocal.
I point out (just to point out the obvious) that this is also true for dividing by a whole number (although we don't do it this way).
So: 10 divided by 2 = 10 times 1/2.
Whew. Nicely done.
I was never gonna get this one.
KerimF - You mean: It is like... tan(65) = 2tan(40) + tan(25)
Be aware of an interesting result when calculating about projectiles.
The math will give you distance, velocity, etc., for any time, but since a projectile thrown up will eventually come back to, say, Earth, we know (ignoring bouncing) that it will then stop. However, the calculation will give you other answers as if there were no Earth to come back to.
A good test for advanced math classes would expect the student to not blindly report the calculated answer, but to also determine when it does not match the physical situation.
An interesting exercise in an algebra class is to treat y (vertical axis) as the independent variable, and define and graph x = f(y).
E.g., parabolas are "sideways," functions must pass the horizontal line test, etc.
If defining functions depending on time, I would definitely use t as the independent (horizontal) variable.
Use whatever makes sense to help keep track of what's going on.
Interesting exercises, but I'd rather you said "use 22/7 for pi" rather than stating it as an equality.
Sure, most folks on this website will understand, but there's always a few who won't.
And then there's always the Indiana Legislature:
https://en.wikipedia.org/wiki/Indiana_pi_bill
A bit sloppy for Khan (or anyone professing to be knowledgeable in this area) to use "velocity" as anything other than a vector.
In common everyday usage, I understand, but not professionally.
That having been said, why don't we call our steering wheel an "accelerator" when we're turning?
And why do we continue to call our gas pedal an accelerator once we have achieved a constant speed?
What is your source please for: "... but I see now that it's a uniform 10^9."
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?