A couple GRE question from a math idiot.

Factual Questions
1

I’m taking the GRE in a couple days. I’ve always been pretty bad at math, and now that I haven’t actually taken a math class in ten years, I’m dreadful. I’ve spent the last few months frantically studying math, but I sometimes run into problems my prep books don’t clarify enough for me. Not sure of what else to do, I turn to you, the Dopers. I have two questions I cannot figure out for the life of me.

  1. One of my math prep books states that squaring a negative number not inside parenthesis results in a negative number. So, (-2)²=4, but -2²=-4. This is NOT what I remember from math class, but, like I said, it’s been a long time and even when I was actually taking math, I sucked at it. My OTHER GRE prep book, however, doesn’t say that at all, and in fact includes a practice question that only makes sense if negative numbers outside of parentheses square into positive numbers.

My books are contradicting each other! What should I believe?

  1. I’ve also got this practice CD-ROM that ETS sent me. There’s a counting principle question in there that just completely baffles me, no matter how many times I read and reread the counting principle section of my prep book. The question is:

You must pick a five person team from a group of twelve people. After picking three members of the team, how many possibile combinations can be made with the last two picks? [This is from memory, so the wording might not sound like an ETS question, but trust me, I have it memorized.]

AFAICT, the number ought to be 72, because with the fourth pick you have 9 possibilites, and with the fifth pick, you have 8. 9x8=72, fin. But the CD-ROM says that the answer is 36 and I CANNOT FIGURE OUT WHY.

Thanks, Dopers!

2

1: With (-2)[sup]2[/sup], the parentheses indicate that you are squaring the whole “-2” thing. In the “naked” case, the question is whether you are putting a negative sign on “2[sup]2[/sup]” or you are putting “[sup]2[/sup]” on “-2”. While you’ll find the occasional exception, the first interpretation of “-2[sup]2[/sup]” is more standard. That is, exponents act before negation, giving: -2[sup]2[/sup]=-4.

These conventions are called, collectively, the “order of operations”. This particular issue is treated in the fifth example here.

2: You have the right idea, but 72 would be correct only if picking “Sally, then Bob” is different from “Bob, then Sally”. Since these two teams (both of which you counted in your 9x8=72) are the same, you need to divide by 2. More generally, you need to divide by the number of different ways you could have picked the two remaining people: 2x1 (same idea that got you to 9x8). Compare “permutation” and “combination” (on the web or in your prep books.)

Hope this helps!

3

Pasta, I REALLY appreciate that.

I’m just kind of worried now because of this problem from my Barron’s test book. This is one of those A-Column A is larger/B-Column B is larger/C-they’re equal/D-impossible to tell questions.

a= -2²

column A
a⁴-a³+a²-a

column B
a-a²+a³-a⁴

According to the book, Column A equals 30, and Column B equals -30, so A is bigger. Now, it’s always a possibility that I’m doing this wrong, but I did it several times, and the only time I got the same answer as the book was when I changed the signs of the superscripted numbers. (ie, if I wrote a⁴ as -2⁴=16 instead of -16…does this make sense?) If you write -2⁴ = -16, Column A equals -20 and Column B equals 20, and B is the answer.

This is probably looking pretty pedantic and foolish, but I’m getting really anxious about this test and can’t figure out if it’s me or the book that is fucked up. (It’s probably me, but I can’t see where I made my mistake.)

4

I don’t know how you keyed in those characters, or whether you see it correctly on your machine, but your Column A starts with “a” followed by a little square, instead of a math symbol.

5

You’ve written something wrong because column A is 340 and B is -340. That doesn’t really matter though. They’re not testing you on your arithmetic. Notice that the signs on each term are the opposite from A to B (a^4 becomes -a^4.) This tells you that one column is positive and the other is negative (or both zero.) If you go through column A and check whether each term is positive or negative, you’ll find that they’re all positive, so it has to be larger than B.

6

Ditto.

But I’ll offer a suggestion that might address the problem.

An expression like a – a[sup]2[/sup] involves squaring whatever number a stands for. It’s equivalent to a – a*a. So if a is a negative number, that negative number will get squared (multiplied by itself). Even though there are no parentheses in the expression, you sometimes have to put them in when replacing variables with numbers (in fact, when in doubt, always put them in). So if a is –7, a[sup]2[/sup] is (–7)[sup]2[/sup], not –7[sup]2[/sup], because whatever number a is, that’s what you have to square.

And then, in a – a[sup]2[/sup], you’d subtract a[sup]2[/sup] from a. So, again if a = –7, a – a[sup]2[/sup] = (–7) – (–7)[sup]2[/sup] = –7 – 49 = –56.

7

That makes sense! Thank you so much.

I don’t know why my superscript 4 is showing up as a box for some of you…I’m seeing it as a 4.

Seriously, people, thank you. I really appreciate it.

8

For what it’s worth, I’m seeing it as a superscript 4 now that I’m at work, but at home I saw it as a box. Maybe it’s because of having different fonts loaded on the different computers?

9

To expand on what Thudlow Boink said, an example:

If a=4 and I ask what 2a is, obviously it is 8. If a=3+6 and I ask what 2a is, I’m asking what twice ‘a’ is, so twice 9, which is 18. Notice that if I blindly write characters down on a piece of paper, I might write:

2a = 2 x 3 + 6

having substituted in a=3+6. But in doing so, I’ve introduced a bad ambiguity – do I want “6 more than 2x3” or do I want “twice 3+6”. The original request was clear, though: 2a means I want double whatever ‘a’ is. If explicitly making the substitution in written form, I can keep the original request clear with parentheses, which say “evaluate the insides of these first”:

2a = 2 x (3 + 6)

The set of conventions called the “order of operations” are in place to keep people from having to write parentheses all the time to specify the order they want. In the above example, multiplication is assumed to act before addition, so 2x3+6=6+6=12, which is fine, but it’s not what was originally asked: 2a=2x(3+6).

So then a[sup]2[/sup] wants “the square of whatever a is”. If a=3 and I choose to explicitly plug in for ‘a’, there’s no confusion:

a[sup]2[/sup]=3[sup]2[/sup]=9

but if a=-6 and I write:

= -6[sup]2[/sup]

people will (by convention) assume I want -36 when I really wanted:

= (-6)[sup]2[/sup].

It this was a point of confusion (it may not have been), then a final thought: Think about variables not as text strings. Think about them as having a single numeric value. If that value happens to be writable as a string of characters, great, but the variable itself is equal to the number that that string of characters evaluates to, rather than the string of characters itself. A silly example would be:

What’s 2h if h=“your age in years”.

‘h’ clearly doesn’t equal the phrase “your age in years”… it equals, say, 114. (Are you old?) Clearly 2h doesn’t equal:

= 2your age in years <-- character replacement

but rather 2h=2x114 = 228. Same thing holds if character replacement would have resulted in a non-silly expression, like a[sup]2[/sup]=-6[sup]2[/sup].