July 4Th Junenth 1865 Because My Ancestors Weren’t Free In 1776 shirt

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Think of -1 as switching directions on the July 4Th Junenth 1865 Because My Ancestors Weren’t Free In 1776 shirt but I will buy this shirt and I will love this number line or turning around 180°, now when you multiply two negative numbers (-a)•(-b) = (-1)•(-1)•a•b you essentially turn around twice and thus turn a full 360°, meaning (-1)•(-1)=1 In short, if we want to keep the properties that multiplication distributes over sums; that 0 times anything is 0 and; that 1 times anything is that thing, then we must have that (-1)(-1)=1. You should probably mention that 0 times anything is 0 is not a trivial fact. You can show that property using distributivity and the existence of an additive inverse. It really depends on what you assume. If we’re just assuming the usual addition and multiplication operation over the naturals, multiplication is usually defined inductively as such:

July 4Th Junenth 1865 Because My Ancestors Weren’t Free In 1776 shirt, hoodie, tank top, sweater and long sleeve t-shirt

In order to derive 0x=0 for all x from distribution of multiplication over addition you need to be working with rings (since, then, you can just assert the July 4Th Junenth 1865 Because My Ancestors Weren’t Free In 1776 shirt but I will buy this shirt and I will love this existence of additive inverses). If, however, you already know that you’re working with a ring, then (-1)(-1) is 1 automatically. For any abelian group G, let ng denote gn (using multiplicative notation even tho it’s abelian just to differentiate the action from the exponent of g), for every integer n and every element g in your group. Then, (-1)(-1) just means “the additive inverse of (-1)” – which is 1. So if you’re defining Z as the groupification of N, and defining multiplication via Z action, then yeah (-1)(-1)=1 is almost an axiom. However, if you’re building Z from N as the set of equivalence classes of NxN modulo (a, b)~(c, d) iff a+d=b+c, then you have your work cut out for you to either prove that such a set is an abelian group (and then get (-1)(-1)=1 for free) or prove it directly (which is what I did).

1. Length is measured from the highest point on the collar down to the bottom hem. 2. Width is measured across the body of the shirt under the armpits, one way. 3. Sleeves are measured from center back to hem.