Math question

[Sharilynn29]

If you saw -4², would your answer be 16 or -16? Why? What about (-4²)? What about -(4²)?

[8filltheheart]

The answer would be -16. The - only belongs to the square if placed inside parantheses. ETA: I should stipulate that the square is outside of the parantheses as in (-4)^2. All of the examples you posted are -.

[regentrude]

If you saw -4², would your answer be 16 or -16? Why? What about (-4²)? What about -(4²)?

 

-4² = - 16. Exponent comes before multiplication by minus one.

(-4²) = -16, same as above; the parentheses OUTSIDE the expression do not change anything.

-(4²) = -16 a well. The parentheses are unnecessary, but makes it clearer what is going on.

 

Only (-4)² = +16, because the " -" would be squared as well if it is inside the parentheses.

[KAR120C]

If you saw -4², would your answer be 16 or -16? Why? What about (-4²)? What about -(4²)?

The "-" is interpreted as multiplication (times -1), so the exponent happens first. The way to make it happen before the exponent would be to put only the -4 in the parentheses, as (-4)².

[Sharilynn29]

Thanks for the explanation. I forgot about order of operations. Dub.:tongue_smilie:

[jessicalb]

Thanks for this thread. Somehow I missed this concept in all the mathing I have done!

[mathwonk]

i agree with 8filltheheart, but don't worry about this kind of thing. I am a PhD in math (1970) and have taught college since 1965 and do not really know the answer to this completely inconsequential question.

 

when faced with this question i would just ask, or look in an algebra book like that of euler. your i.q. does not go down if you have no clue as to the answer to this question ( i hope).:001_smile:

 

relax, math is not about questions like this.

The "-" is interpreted as multiplication (times -1), so the exponent happens first. The way to make it happen before the exponent would be to put only the -4 in the parentheses, as (-4)².

 

But why is -4 treated as a compound operation (-1 * 4), and not a single conceptual number? I looked it up, and it is correct, but just seems confusing. I think we'd all agree that 12 squared is 144, even though 12 is really 1 * 10 + 2 -- no one would say that 12 squared is 14 (i.e 1 * 10 + 2 squared) , but that seems to be the same thing as ripping the negative sign off of the 4, and applying the exponentation operator to the absolute value of the scalar -4.

[creekland]

Having an established order of operations is important so that people everywhere can agree upon an answer. Otherwise, you'd end up with the confusion you see here when different people look at things in different ways. ;)

 

We DO teach the order of operations in school and -4^2 is -16 whereas (-4)^2 is 16. It is important to know in our (public) school. ;)

 

Consider something like 10 - 4 + 1.

 

The answer could be 10 - 4 = 6, then + 1 = 7 (this is correct BTW).

 

OR, many who do math in their heads could see 4+1 = 5, then 10 - 5 = 5 (incorrect --> addition and subtraction go left to right).

 

By having established rules, we know what the math world agrees upon as correct. If we didn't, some problems would create chaos.

[KAR120C]

But why is -4 treated as a compound operation (-1 * 4), and not a single conceptual number? I looked it up, and it is correct, but just seems confusing. I think we'd all agree that 12 squared is 144, even though 12 is really 1 * 10 + 2 -- no one would say that 12 squared is 14 (i.e 1 * 10 + 2 squared) , but that seems to be the same thing as ripping the negative sign off of the 4, and applying the exponentation operator to the absolute value of the scalar -4.

We do tend to think of a -4 as "negative four" and a single value... but as Creekland says it's a matter of having a standard that we can be consistent with. In this case I think it's part of the problem with using the same sign for negative as for minus. If you had 1-4², it's clear (or at least clearer) that the four is the value that's squared and the operation in front of it is subtraction and happens after... or if you had -1*4² it wouldn't be confusing. It's only when you have the negative and the squaring run together as -4².

 

This sort of thing really is kind of a trick question. Not because order of operations isn't absolutely vital, but because this particular point of it will catch so many people that if you want to be sure of communicating clearly, I'd use the parentheses.

[kiana]

But why is -4 treated as a compound operation (-1 * 4), and not a single conceptual number? I looked it up, and it is correct, but just seems confusing. I think we'd all agree that 12 squared is 144, even though 12 is really 1 * 10 + 2 -- no one would say that 12 squared is 14 (i.e 1 * 10 + 2 squared) , but that seems to be the same thing as ripping the negative sign off of the 4, and applying the exponentation operator to the absolute value of the scalar -4.

 

Either way we went, we'd end up with cases that seem ambiguous. If -4² gave the result of 16, then what about 0 - 4²? Would that give you 0 (since the - would be attached to the 4, we wouldn't have an operator, and juxtaposition indicates multiplication), 16 (treating -4² as +16), or what? In order to make it clear, we'd have to say 0 - (4²).

 

Furthermore, this would also mean that something like 3x² was ambiguous and could mean either 3(x²) or (3x)². In order to make this unambiguous, we'd need to write it as 3(x²). This would honestly make writing polynomials a nightmare.

We do tend to think of a -4 as "negative four" and a single value... but as Creekland says it's a matter of having a standard that we can be consistent with. In this case I think it's part of the problem with using the same sign for negative as for minus. If you had 1-4², it's clear (or at least clearer) that the four is the value that's squared and the operation in front of it is subtraction and happens after... or if you had -1*4² it wouldn't be confusing. It's only when you have the negative and the squaring run together as -4².

 

I completely understand order of operations and why it is important, but I still don't understand why when you just have a -4, why the negative sign is considered an operation, and not intrinsically part of the number. If you have 1 - 4², to me, there isn't a negative four there, there is a subtraction operation on the result of 4 squared. What if you had 1 - -4²?

[css3238]

I completely understand order of operations and why it is important, but I still don't understand why when you just have a -4, why the negative sign is considered an operation, and not intrinsically part of the number. If you have 1 - 4², to me, there isn't a negative four there, there is a subtraction operation on the result of 4 squared. What if you had 1 - -4²?

 

Try changing it into words and maybe that will help. -4 should be properly read "the negative of four," not "negative four." You have to remember that numbers are merely symbols that represent an amount of something. There is no such thing as "negative four" because there is no such thing as "four." There are "four apples" and "four hats" but there is no "4," it is a symbol we made up.

 

Once you see it that way, you begin to understand that there cannot be "negative four apples" or "negative four hats." We can owe someone four apples or hats, but that does not create a situation where there are negative apples or hats. It just isn't possible. So we represent that owing as the negative of the amount, or -4 in this case.

 

I took a long way to get here, but here's the pay off. If you add the square to the "proper" reading of the number, it becomes "the negative of four squared." That explains why the answer is -16. At least I hope that makes sense.