## -1^2: is this equal to -1 or to 1?

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### -1^2: is this equal to -1 or to 1?

Ok, I did a little reading on this and thought I had the concept down but apparently I'm missing something. The big question is does$-1^2$ equal -1 or 1?
It's -1 from the explanation I read which makes perfect sense to me, if $-1^2 = (-1^2)$ =$-(1)(1)$ then$-1^2$ = $-1$. If, $(-1)^2$ this is (-1)(-1)= 1 So fine, the negative is applied to the exponent in the first example and not in the second.
The problem I'm having a little trouble with is:
$4y^2-3 = 1.$ with the question: Is -1 a solution?
I say it's:
Substituting -1 for y
$4(-1^2)-3 = 1$
$4(-1) - 3 = 1$
$-4 - 3 = 1$
So negative one is not a solution.
$4y^2-3 = 1$
$4y^2 = 4$
$y^2 = \frac44$ or $y^2 = \frac14(4)$
$y^2 = 1$
y = 1

Um, I think....
diaste

Posts: 14
Joined: Sat Jan 17, 2009 1:54 pm

### Re: -1^2

diaste wrote:The problem I'm having a little trouble with is:
$4y^2-1 = 5.$ with the question: Is -1 a solution?
I say it's:
Substituting -1 for y
$4(-1^2)-1 = 5$
$4(-1) - 1 = 5$
$-4 - 1 = 5$
So negative one is not a solution.

You have the right conclusion, but for the wrong reason.

To see why, add parentheses around the variable first, before you substitute in the -1:

$\begin{eqnarray}
4y^2\,-\,1\, \overset?=\,5 \\
4(y)^2\,-\,1\, \overset?=\,5 \\
4(-1)^2\,-\,1\, \overset?=\, 5 \\
4(1) \,-\, 1\, \overset?=\, 5 \\
4 \,-\, 1\, \overset?=\, 5 \\
3 \,\neq\, 5 \\
\end{eqnarray}$

$4y^2-1 = 5$
$4y^2 = 6$
$y^2 = \frac64$ or $y^2 = \frac14(6)$
$y^2 = 1.5$
y ~ 1.22

Um, I think....

Yes, that's one solution. Remember, though, there are two solutions when you're taking the square root of a positive number.

DAiv
DAiv

Posts: 44
Joined: Tue Dec 16, 2008 7:47 pm

### Re: -1^2

I think I see it now. The substitution is -1 not $-1^2$. obviously the square is then $(-1)^2$ and not $(-1^2)$.

There are two solutions when taking the square root of a positive number? Really? What's the other solution?

Thanks again!

Daniel
diaste

Posts: 14
Joined: Sat Jan 17, 2009 1:54 pm

### Re: -1^2

diaste wrote:There are two solutions when taking the square root of a positive number? Really? What's the other solution?

It's probably easiest to explain with an example:

$\begin{eqnarray}
y^2 \,&=&\, 4 \\
y \,&=&\, \pm \sqrt 4 \\
y \,&=&\, 2 \text{ or } y \,=\, -2
\end{eqnarray}$

Check:

$(2)^2 = 4 \text{ and } (-2)^2 = 4$

DAiv
DAiv

Posts: 44
Joined: Tue Dec 16, 2008 7:47 pm

### Re: -1^2: is this equal to -1 or to 1?

For -1^2, the 2 is only on the 1, not the -. If you had (-1)^2, then the 2 would be on the -.
anonmeans

Posts: 50
Joined: Sat Jan 24, 2009 7:18 pm

### Re: -1^2: is this equal to -1 or to 1?

anonmeans wrote:For -1^2, the 2 is only on the 1, not the -. If you had (-1)^2, then the 2 would be on the -.

This is true for pure mathematics. It's worth noting, however, that the Microsoft Excel spreadsheet (and some computer languages) gives the unary minus sign precedence over the exponent, so -1^2 is actually interpreted as (-1)^2. There, explicit parentheses are needed to get the desired result, i.e. -(1^2). Just something to be aware of if anyone uses that product.

DAiv
DAiv

Posts: 44
Joined: Tue Dec 16, 2008 7:47 pm

### Re: -1^2: is this equal to -1 or to 1?

So Excel and stuff doesn't do the order of operations? That's confusing!

little_dragon

Posts: 188
Joined: Mon Dec 08, 2008 5:18 pm

### Re: -1^2: is this equal to -1 or to 1?

Anything squared is itself times itself. (-1) squared is (-1)(-1). Since a negative times a negative is a positive, (-1)(-1) is 1.
mathgirl

Posts: 1
Joined: Tue Jan 27, 2009 5:23 pm

### Re: -1^2: is this equal to -1 or to 1?

little_dragon wrote:So Excel and stuff doesn't do the order of operations? That's confusing!

It does all the other operations in the correct order, but it swaps the order of the unary minus sign (i.e. the minus in '-5') and exponent (i.e. the 2 in x^2).

So, pure maths has the order:
parentheses,
exponent,
unary minus sign | multiplication | division,
addition | subtraction (i.e. binary minus)

and Excel has the order:
parentheses,
unary minus sign,
exponent,
multiplication | division,
addition | subtraction (i.e. binary minus).

It's important to distinguish between unary minus and binary minus, though. A unary minus operates on just one thing (operand), e.g -5, while a binary minus operates on two operands, e.g. 7 - 5.

So, in pure maths:
-5^2 = -(5^2) = -(25) = -25, because the exponent '^2' is performed before the minus sign '-'

but in Excel:
-5^2 = (-5)^2 = 25, because the minus sign '-' is peformed before the exponent '^2'

And you can also have both a binary minus and a unary minus together, as in:
6 - -2 = (6) - (-2) = 6 + 2 = 8

where the first minus is binary (as it has the two operands '6' and '-2') and the second one is unary (as it has just the one operand '2').

DAiv
DAiv

Posts: 44
Joined: Tue Dec 16, 2008 7:47 pm