What is -1 squared

Question

I am stuck on this question, some people say it is 1 and the others say it's -1

Answer 1

Squaring is multiplying the number twice, so that means:

-1 * -1

A negative times a negative equals a positive, and 1 times 1 equals 1, so -1² is 1.

 

Answer 2

I'm not sure what level you're asking this question at;

 

Rules of integer arithmetic to consider:

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additive identity:  0 + a = a, for all a

multiplicative identity: 1a = a1 = a, for all a

distributive rule for multiplication: a(b+c) = ab + ac, for all a,b,c

associative law for addition: (a+b)+c = a+(b+c), for all a,b,c (there is also an associative law of multiplication that we don't need here)

additive inverse: for all a there is a value called -a such that a + -a = 0

 

These are called axioms, and are developed in something called abstract algebra (ring theory); it is itself related to a subject of the last century called math foundations. However note that the question's answer is not an axiom; we are having to prove that (-1)*(-1)=1 using the axioms above that we simply accept as given.

 

Now it is simple:

First we show that 1*0 = 0

1*(0+a)=1*a=a (by additive identity)

=1*0 + 1*a (by the distributive rule)

=1*0 + a (by multiplicative identity)

We now use the additive inverse and associative law to conclude

a = 1*0 + a

a + (-a) = 0 = 1*0 + a + (-a) = 1*0

 

Using the fact just shown, and the rules above (you can fill them in), we have:

 

-1 * (1 + -1) = -1*0 = 0

-1 * 1 + (-1) * (-1) = 0

-1 + (-1) * (-1) = 0

-1 + (-1)*(-1) + 1 = 1

(-1)*(-1) = 1