Math question- What is a negative number squared?

[ready123go]

Thanks for all of the explanations! I understand more what the teacher is saying now. I really don't think that is how I was taught so it is going to take some adjustment in my thinking when helping him.

Going with what the teacher is saying is a sound & solid plan!

Frankly, I began to doubt myself and searched both Google (with varying results) and Wikipedia.

Wikipedia has a good (if complex) explanation and shows why there is more than one camp (and a why negative integer has an imaginary square root). Square of a negative number

 

[burnurcomputer]

And its things like this that are supposed to be taught in the common core, maybe it will end up a positive thing in the long run?

*Not trying to stir the pot, but just trying to show that this is not understood very well, even by myself)

 

[BCDisneyFanatic]

But 'squaring' a number means multiplying it by itself, right?

So 7^2 is the same as 7 * 7, right?

So if -7 is squared, -7 multiplied by itself, so -7 * -7.

For those who say the negative sign just indicates multiplying by -1, then why isn't -7^2 also written as (-1 * 7) * (-1 * 7)?

Is -7 (not squared, just the number) an integer or a calculation?

I really think the problem with this problem is using the spoken word to determine an equation.

-7 is an integer; however, -7^2 is an expression, not an integer. The negative sign is a coefficient of -1 in front of 7^2, in the same way it is assumed by convention that there is a coefficient of -1 in front of -x. To simplify -7^2 (and find the integer the expression represents), you must follow the order of operations...and exponents come first before multiplying coefficients (if there are no brackets!).

 

[Cannot_Wait_4Disney]

I feel like an idiot for asking this question but I don't know if I was taught wrong in school. However, if I was taught incorrectly several of my friends were as well because they agree with me.
I have always thought a negative number squared is a positive number such as a -7 squared (I don't know how to make the little two on my ipad) is 49. However, my son's teacher is saying that the answer is -49 as you do the exponent portion first (7x7) then you add back the negative sign after. What?? She states the only way it is positive is if the -7 is in parentheis.
Are there any math wizards out there that can help me?

Any negative number squared is a positive number. But you have to be very careful of notation. And you also have to be very careful of translating what is said in English to notation.

For some reason I can't notate an exponent here like I normally do.

(-7) squared or (-7)^2 is how to notate that you wish to square negative 7. This would be 49.
-7squared oe -7^2 is how to notate that you wish to take the negative of 7 squared. This would be -49

The big problem is people tend to get loose with their notation and loose with translating English to notational form and hence the confusion. Some books don't even get it right all the time.

 

[sam_gordon]

-7 is an integer; however, -7^2 is an expression, not an integer. The negative sign is a coefficient of -1 in front of 7^2, in the same way it is assumed by convention that there is a coefficient of -1 in front of -x. To simplify -7^2 (and find the integer the expression represents), you must follow the order of operations...and exponents come first before multiplying coefficients (if there are no brackets!).

I understand the argument. I just disagree. And maybe it's because I'm going back to "old" math.

I agree you can rewrite -7^2 as -1 * 7^2. However, IMO, *IF* -7 is an integer, then it's an integer always and should be treated as one when doing the order of operations.

I haven't taken a math class in a long time, and this is simply my opinion.

 

[Cannot_Wait_4Disney]

For those who say the negative sign just indicates multiplying by -1, then why isn't -7^2 also written as (-1 * 7) * (-1 * 7)?

-7^2= -1(7*7)

The distributive property cannot be used on -1(7*7) to get (-1*7)*(-1*7)
The former is -49, the latter 49.

The distributive property
x(y+z)= x*y+x*z= xy+xz.

Cannot be applied to the general case of
x(y*z) to get (x*y)*(x*z) The former is xyz the latter is X^2yz.

or in the case where x=-1,y=7, and z=7 the former yields -49, the latter 49.

 

[sam_gordon]

-7^2= -1(7*7)

The distributive property cannot be used on -1(7*7) to get (-1*7)*(-1*7)
The former is -49, the latter 49.

The distributive property
x(y+z)= x*y+x*z= xy+xz.

Cannot be applied to the general case of
x(y*z) to get (x*y)*(x*z) The former is xyz the latter is X^2yz.

or in the case where x=-1,y=7, and z=7 the former yields -49, the latter 49.

I'm not using the distributive property. I'm calling -7 an integer. So,
-7^2 = -7*-7.

If you want to expand it some more...

-7 * -7 = (-1*7) * (-1*7).

According to http://www.mathsisfun.com/definitions/square-number.html, the definition of a square number is

The number you get when you multiply an integer by itself.

If -7 is an integer, then you multiply it by itself to square it and you get 49.

-7 is a specific number (integer) and not an operation. Again, my opinion.

 

[sam_gordon]

ooh, just thought of something.

We've been discussing this as the equation -x^2, with x=7. However, IS that the proper equation?

What if you make the equation x^2, x=-7?

 

[SandrA9810]

I understand the argument. I just disagree. And maybe it's because I'm going back to "old" math.

I agree you can rewrite -7^2 as -1 * 7^2. However, IMO, *IF* -7 is an integer, then it's an integer always and should be treated as one when doing the order of operations.

I haven't taken a math class in a long time, and this is simply my opinion.

But your old school math isn't going to help some one pass a math class.

 

[SandrA9810]

ooh, just thought of something.

We've been discussing this as the equation -x^2, with x=7. However, IS that the proper equation?

What if you make the equation x^2, x=-7?

If x is equal to -7, then you would put it into brackets. That's the defining difference.

Parenthesis, exponents, multiply, divide, add, subtract. We all know the PEDMAS rule.

(-7)^2 is the same as (-1*7)^2. The power of two belongs to both the one and the seven. -1 becomes positive 1, and 7 squared is 49. 1*49=49
-7^2 is the same as -1*7^2. 7*7=49 is step one, exponents before multiplication, -1*49= -49

Anytime you're solving for a variable, then you put your answer into parenthesis when solving the equation to check the answer.

So if x=9, y=-2, z=-4

z(x*y) = (-4)*[(9)*(-2)]

 

[BCDisneyFanatic]

I'm not using the distributive property. I'm calling -7 an integer. So,
-7^2 = -7*-7.

If you want to expand it some more...

-7 * -7 = (-1*7) * (-1*7).

According to http://www.mathsisfun.com/definitions/square-number.html, the definition of a square number is

If -7 is an integer, then you multiply it by itself to square it and you get 49.

-7 is a specific number (integer) and not an operation. Again, my opinion.

Technically, -7 can be both an integer and an expression of -1 x 7. That is why, to avoid all this confusion, mathematical minds much greater than mine have put rules into place!

If you want to indicate that you are squaring the integer -7, then it MUST be put into brackets; otherwise, other mathematicians will assume by convention that you are only squaring the 7.

Yes, -7 x -7 = +49. However, in exponential form, that needs to be written (by convention) as (-7)^2.

I didn't make the rules...it is just important that everyone follows the same conventions so that they get consistent answers.

 

[pkondz]

-7^2 = -49 = my calculator's broken 'cause it says 49.

 

[Cannot_Wait_4Disney]

I'm not using the distributive property. I'm calling -7 an integer. So,
-7^2 = -7*-7.

If you want to expand it some more...

-7 * -7 = (-1*7) * (-1*7).

According to http://www.mathsisfun.com/definitions/square-number.html, the definition of a square number is

If -7 is an integer, then you multiply it by itself to square it and you get 49.

-7 is a specific number (integer) and not an operation. Again, my opinion.

You are entitled to your own opinion, but I am entitled to point out your errors. .

Taking the integer -7 and squaring it is (-7)^2. That is not equal to -7^2 as I have shown. It can't be any plainer than that. You can either learn that, or you can continue to deny reality. I'll leave that choice to you.

 

[sam_gordon]

you are entitled to your own opinion, but i am entitled to point out your errors. .

Taking the integer -7 and squaring it is (-7)^2. That is not equal to -7^2 as i have shown. It can't be any plainer than that. You can either learn that, or you can continue to deny reality. I'll leave that choice to you.

ok

 

[BCDisneyFanatic]

You are entitled to your own opinion, but I am entitled to point out your errors. .

Taking the integer -7 and squaring it is (-7)^2. That is not equal to -7^2 as I have shown. It can't be any plainer than that. You can either learn that, or you can continue to deny reality. I'll leave that choice to you.

Eh, I don't see the need for the snark. Sam's asked some interesting (and legitimate) questions, and I've enjoyed the mathematical debate.

 

[sam_gordon]

Eh, I don't see the need for the snark. Sam's asked some interesting (and legitimate) questions, and I've enjoyed the mathematical debate.

Thanks for the support BC! I'm genuinely not trying to be difficult, I just want to make sure *I* understand. I texted my DD (who's in pre-AP Alegebra II and pre-AP Geometry) to ask her Alegebra teacher, and the response I got (from her and her AP World History teacher) was "you're insane".

 

[EMAW_KSU]

You are entitled to your own opinion, but I am entitled to point out your errors. .

Taking the integer -7 and squaring it is (-7)^2. That is not equal to -7^2 as I have shown. It can't be any plainer than that. You can either learn that, or you can continue to deny reality. I'll leave that choice to you.

So when you solve a quadratic equation like this, you do not get two correct answers?

x^2-49 = 0

We can rewrite that equation two ways.

(x-7)*(x+7)=0

so (x-7)=0 or (x+7)=0
so x=7 or x=-7

The second way to write the equation is
x^2 = 49

x=7 only?

How can the same equation have different answers?

 

[BCDisneyFanatic]

So when you solve a quadratic equation like this, you do not get two correct answers?

x^2-49 = 0

We can rewrite that equation two ways.

(x-7)*(x+7)=0

so (x-7)=0 or (x+7)=0
so x=7 or x=-7

The second way to write the equation is
x^2 = 49

x=7 only?

How can the same equation have different answers?

Well, technically there are two values of x for which the equation is valid, and the correct 'answer' should include both values.

There is still only one correct value when -7^2 is simplified, though and that is -49 .

(There are two possible solutions to the opposite operation, though...the square root of +49 is either -7 or +7).

 

[SandrA9810]

So when you solve a quadratic equation like this, you do not get two correct answers?

x^2-49 = 0

We can rewrite that equation two ways.

(x-7)*(x+7)=0

so (x-7)=0 or (x+7)=0
so x=7 or x=-7

The second way to write the equation is
x^2 = 49

x=7 only?

How can the same equation have different answers?

You would put your answers in parenthesis when checking the solution, and you will come up with the correct answer. -7 would be inside parenthesis and therefor become a positive number.

 

[sam_gordon]

There is still only one correct value when -7^2 is simplified, though and that is +49 .

Wait, what? I thought you (and others) said -7^2 = -49?