I’m currently studying for the GREs and just about ALL of my basic math skills have gone out the window. I began studying using the Kaplan prep book, but I’ve also just enrolled in a basic math proficiency course at my local university. As far as I can tell, there seems to be a discrepency.
According to Kaplan:
2x[sup]2[/sup] = 32
which is bigger: x OR 4?
The answer is that there is not enough info to tell because it is possible that x = -4
However, the text for the class I’m in says:
-5[sup]2[/sup] = -(5 x 5) = -25
while
(-5)[sup]2[/sup] = (-5)(-5) = 25
Is there a discrepency here? Because I think I would’ve taken it for granted that -5[sup]2[/sup] = 25
“Negating” a number (multiplying it by -1) comes after exponentiation in the order of operations. So when you see, for example, -5[sup]2[/sup], that means “take five, square it, then multiply it by -1”. On the other hand, (-5)[sup]2[/sup] means “take five, multiply it by -1, then square that”. The former gives you -25, the latter gives +25.
Also, when you have, for example, the espression x[sup]2[/sup], and you are given x = -3, then “both” the -1 and 3 get squared–i.e., in this case, we would write x[sup]2[/sup] = (-3)[sup]2[/sup] = +9.
While it is true that -x[sup]2[/sup] is defined as -(x[sup]2[/sup]), I say you are perfectly right to say that the solutions of 2x[sup]2[/sup] = 32 are 4 and -4. This site agrees - that’s why quadratic equations have two roots, right?
So, if I understand you correctly, it’s a dumb question.
IANA Mathematician, but that sure looks like an F-up to me. -5 is identical to (-5), and is not a shorthand way of writing -15 unless the standards have somehow changed in the last few years. Now if the equation was -15[sup]2[/sup], the answer would be -25, since you do the exponent before all other functions. -5[sup]2[/sup] is 25. At least, that’s how it was in MY day.
Somebody was smokin’ doobies when they should have been writing a textbook.
See? This is what happens when you start a reply, go do something else, and come back to it. A slew of responses get posted that may you look like a retard for posting late.
So you guys are saying that while -5*-5=25, -5[sup]2[/sup]=-25?
I’m beginning to remember why math class always filled my with blinding rage, although I still don’t remember it being taught to me that way…
Of course, you still have textbooks teaching estimation differently: Some say 0-4, round down, 5-9 round up. Others (the stupid ones) 0-4 round down, 6-9 round up, 5 round down if the preceeding number is even, up if it is odd.
No, -5[sup]2[/sup] = -25, for reasons stated above. That’s just the way it is.
By the way, when I said it was a dumb question, I didn’t mean the OP, I meant the question about “Which is greater?” Having said that, I think I may have misread, as the answer given (insufficient information) is correct.
However, I think I see where the confusion arises:
Will you feel worse if I point out that the “stupid” way is the right way to do it?
If you go to estimate 1.35 + 1.45, by the first rule, you’d do it as 1.4 + 1.5, and by the second, you’d do it as 1.4 + 1.4. Notice the discrepancy there, and how one gives you the right answer?
The expected error introduced by the second method is 0. The expected error introduced by the first is greater than 0. That suggests that the second method is one heckuva lot better.
I never liked that method (and in fact never saw it in any of my own textbooks) because it skews estimations lower. Given that the last digit of any number has an equal chance of being any digit (0-9), by using what I will maintain until the day I die is the “stupid” method, a number has a 55% chance of being rounded down, whereas the first method it’s flat 50-50.
Example: 2.04=2, but by the second method, so does 2.54.
2.54 rounds to 3 when you’re rounding to the nearest whole integer no matter what method you’re using, because 3 is nearer to 2.54 than 2 is. Most textbooks probably botch the explanation quite badly, but that’s a poor reflection on textbook writers, not mathematicians.
The only case in which 5 rounds toward the even is when the difference between the original number and the even is the same as the difference between the original number and the odd (i.e., when there is no nearest).
And Brane, I feel your pain. Very very much so. I was a top math student back in HS. Took calculus my first semester of college and never looked back. That class, my very last math class, was 11 years ago. I’m scared to death of these GREs.
How have we made it to 16 posts about the fact that -(x)[sup]2[/sup] != (-x)[sup]2[/sup]?
[hijack=“rounding”] |X - (2X+1)/2| = |X+1 - (2X+1)/2|
Rules for rounding in this case (I.E. the first insignificant digit is 5) are arbitrary.
Using the (2X+1)/2 = X+1 rule for rounding (I.E. If the first insignificant digit is: [1,2,3,4], round down; [5,6,7,8,9], round up) introduces positive bias.
Using the (2X+1)/2 = X rule for rounding (I.E. If the first insignificant digit is:[1,2,3,4,5], round down; [6,7,8,9], round up) introduces negative bias.
Using a more complex rule that invokes both cases equally will cause the bias introduced by both methods to cancel. Thus the “If the first insignificant digit is 5 and the least significant digit is: [0,2,4,6,8], round down; [1,3,5,7,9], round up.” should yeild a smaller cummulative error. Worst case scenario (highly sterotypical sets of numbers (I.E. the least significant digit is always odd or always even)) the cummulative error would be equal to either of the simple “Round 5 Up” or “Round 5 Down” rules applied to the same set.
If anyone else feels like running the numbers, feel free.
[/hijack]
If I say x= -5, then x[sup]2[/sup] = 25, because x[sup]2[/sup] means (-5)[sup]2[/sup]
When determining the value of the expression x[sup]2[/sup], you don’t just write -5 in place of x and get -5[sup]2[/sup]. If you did, you’d think x[sup]2[/sup] was -25 (see *), which is wrong. In this case you have to add the parentheses when going from variables to numbers to convey that the minus sign is part of the expression being squared.
*:
-5[sup]2[/sup] means -(5[sup]2[/sup]), rather than (-5)[sup]2[/sup]. It’s confusing, because we tend to read them both as “negative five squared.”
Significant figures and rounding are engineering problems, not mathematical ones. Suppose you have a 2ft by 1.27ft piece of metal that you have to paint. Mathematically, you might be able to say “2.54 sqft!”, but from an engineering standpoint, 2ft is anything between 1.5 to 2.5ft (unless other tolerances are given) and 1.27ft is anything between 1.265 to 1.275ft. Taking the extremes of our tolerances, the area of our piece of metal is anything from 1.8975 to 3.1875sqft. Thus, it is acceptable to round 2.54sqft down to 2sqft, or up to 3sqft.
Shorter version, only the first insignificant digit is, er, significant. In 2.54, 2 is our least significant digit (the one we’ll be rounding), 0.5 if our first insignificant digit (the only non-significant digit that matters), and 0.04 effectively does not exist. Rounding 2.54 to one significant digit is an equivalent case to rounding 2.50 or 2.59 to one significant digit. And since 2.5 is no closer to 2.0 than it is to 3.0… either case is acceptable.
