Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt

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Think of -1 as switching directions on the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt but in fact I 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:

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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 Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt but in fact I 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).
Wow this is really cool! I’m a novice with math proofs and only did for engineering. What kind of proof is this? It’s just a straight-up direct proof. You start with a statement you know to be true. Then you follow a series of steps that are logically sound. Then you arrive at your conclusion. The trick is knowing the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt but in fact I love this right “clever” step to stick in the middle. Things like adding 0 to something by using (+1-1) or multiplying by 1 by using some fraction that cancels out are common tricks to achieve this. If you liked this proof you might like more advanced algebra! This proof might feel pretty basic, but the concepts used in this proof(0a=0, 1a=a, associativity, commutativity and distributivity) are essential to grasp there Removing 36 dollars of debt once (-36) x (-1) is the same as straight up giving you 36 dollars. Removing it twice, thrice or six times etc is the same principle.

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Removing 36 dollars of debt should be -36 * 0. That’s “removing the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this debt.” If you do -36 * -1 you’re inverting the debt. Not only is my debt gone, but I’m sitting on an extra $36 now. It’s $72 different from where I was, not $36 different. If I’m $36 in debt and you give me $36, my net is zero. The comparable operation would be something more like this: We’re not saying, the balance started at -36, then we remove 36 dollars of debt, so new balance = -36 * -1 = 36. That’s not how this works. That doesn’t really work. If I’m 36 Dollar in debt and you give me 36 dollar I’m at 0 dollar. But (-36)x(-1)=+36. You are +36$ compared to your previous state. It describes how removing debt equals to giving money in this case. Consider a number line. You were at the position -36, then someone took away all the debt, bringing you to the position 0. You’ve moved 36 units to the right. You are not, you only received 36$. The transaction can be desribed as -36$ (a debt) and -(-36$) meaning you removed the debt. The result is you got 36 dollars. (So you are at 0$, but compared to -36, you have 36$ more) 6 × 6 means you are adding 6 six times, which gives you 36. 6 x -6 means you are adding -6 six times, which also means you are subtracting 6 six times, which gives you -36.
By the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this associative property, which says that in a product we can multiply the numbers together in any order we want, let’s group the product (-1 * -1) together first. We then have: Which we know by our previous proof that -1 * -1 = -(-1). We’re in luck, because our first definition also says that -(-1) is the number that when added to -1 gives 0. Of course, we all know that this number is just 1, and we also know that this number is unique. Therefore, We can note that there is nothing special about 6 in particular; indeed, it is true that -x * -y = x * y for any real numbers x and y. I don’t think that’s the associative property. You didn’t just move parenthesis by grouping them differently, you swapped the order the factors, which is the commutative property
I think they were talking about this line ^^, when they said associative, no? Multiplying (-1*-1) first is a use of the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this associative property. You’re right they probably were talking about the next step, not the previous one. That’s very weird for two reasons though: – if they first wrote the products without parenthesis, it means that they already used the associative property to signal that there’s no ambiguity in how to group it. Either that, or we are using implicit PEMDAS and the first product (-1)(-1) is already the thing you have to compute first regardless. – feels weird to swap (-1) and 6 in the middle of a product of 4 factors without commenting anything as it if it were obvious (it is), only to then proceed to give a detailed explanation of the associative property for why we compute the first product first. If we are operating at this level of “rigor” it seems like the first one warranted an explanation more than the second one
yep, my guess is that this was meant to be a very quick way to show that it can be derived from the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this basic properties of the reals. I actually didn’t bother to check what was done at every step but If we wanted a formal proof that emphasises which axioms we used at every step then yeah this whole thing would need a rewrite. And we’d also probably need to start with proving a*(-b) = – (a*b), as well as 0*b = 0 (depending on how you go about it) then move on to proving the main result. But yeah your last point makes sense, and I’d even go to say as I mentioned above that probably every step should be clarified if we want to be as careful and clear as possible. Honestly this is a fair criticism, but I was operating under the assumption that they already knew what the commutative property was. Also, I was putting more emphasis on the part where we actually show that the product of two negative numbers is positive, and keep the comment less bloated.
Lets take the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this real numbers, where you can add, subtract, multiply and divide. There is a number called “identity for multiplication” such that you multiply this number to x and you just got x. This is one (x * 1 = x). Multiplication also has an “inverse” operation, which is division. A number x divided by itself will get you the identity (x/x = 1). Now, this is going to sound like a silly question, but as you said, this is a foundational issue, so to give a foundational example, I need a really basic and foundational question. What if we ask ourselves, how much money do I need to pay, in order to eliminate my debt to Alice and her friends? Now, obviously you need to earn $100, but can we work that out with maths, rather than pure intution? Is there some core and important principle here?
Actually it is based on principles of real number system. I suggest you to check the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this arithmetics of the the wikipedia page for the real number so you can prove it. I try prove it here so you have some ideas: I answered a similar question yesterday. The short simplified answer is that if you accept the basic axioms and properties of the real numbers like, for example, the distributive property, which allows you to do things like 2*(3+5) = 2*3 = 2*5, then must also accept that -a * -b = a * b, where a,b are real numbers, as this can result can be derived starting with these properties. This answer doesn’t offer a satisfying intuitive explanation so If you would like to see a bunch of intuitive explanations and a few formal proofs for this result, check this out:
– is negation, for example: he is not not a good guy, which means he is a good guy, see ? he is a good guy = he is not not a good guy Has not this question just come up? Well. This is because you wish some basic properties of multiplication and addition to be true. Now, consider (-a)×(-b). Well, (-a)×(-b) = (-a)×(-b) + 0. But a×(-b + b) = 0 because -b + b = 0 and a×0 = 0. So we can say Note that this follows only from the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this laws I gave above. In particular you do not need to assume that a×b = b×a or that a+b=b+a, or that there is a multiplicative inverse. So this thing is true not just for numbers: it is true for things which only have some of the properties of numbers. Hello linear algebra. I was about to make the proof myself but I thought somebody will come up with this. You just want the part with neutral and inverse element to work, the strange multiplication rule is born out of that Another way to see it is to consider the -6 times table: should equal 6, right? And -2 x -6 should equal 12. Then the pattern would be consistent. It would go 12, 6, 0, -6, -12, … (-x)•(-x) = (-1)•(-1)•x•x. However (-1)•(-1) = 1. You think its -1 but if we assume that -1•-1 = -1, than, dividing both sides by -1 youd get -1 = -1/-1. -1/-1 is however 1 because youre dividing a number by the same number. So you get -1 = 1 which is a contradiction
Where cos(x) is the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this real part, and sin(x) is the imaginary part (becuase it is multiplied by i which is not a real number) Think it this way: You have no money, and you are not have no money., that means you have money. And multiply it with a number means how much you have or you don’t have, magnitude. Does it make sense to you now? Distributivity is the reason we need Think of a graph – 6×6 is the area enclosed from 0,0 to 6,6 (opposite points of a rectangle). -6 x -6 is the area enclosed from 0,0 to -6,-6. Both are 36. The rational, real and complex numbers are what’s called fields. In abstract algebra, a field is a structure with “addition”, “subtraction”, “multiplication” and “division” operations like the ones we’re used to. Formally, the definition is: There is a binary operation which we call “addition” and whose result we denote as a+b, and a binary operation which we call “multiplication” and whose result we denote ab. The operations are closed: for any pair of field members a and b, the results a+b and ab are also members of the field. Each field member has an additive inverse: for every a there exists a field member -a such that a + (-a) = 0. We may also notate a + (-b) as a – b, which defines a subtraction operation.
Each field member except 0 has a multiplicative inverse: for every a except 0 there exists a field member a-1 such that aa-1 = 0. We may also notate ab-1 as a / b, which defines a division operation. Finally, the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this multiplication operation is distributive over the addition operation: a(b+c) = ab + ac for all field members a, b, c. Now, using only these properties we can show that (-a)(-a) = aa must be true for any number system that has these properties. Let’s begin by proving that 0a = 0: This uses property (5) to make the substitution 0 = 0 + 0, and then property (9) for the second step. Subtracting 0a from both sides (properties (3) and (6)) we get: This uses property (7) to substitute a = 1a, property (9) for the second step, the proof above to substitute 0a = 0, and finally property (6) for the last step. Now subtract a from both sides (properties (3), (4) and (6)): So now we know that we can write
To summarize: if you’ve been keeping track, this all depended solely on the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this properties (3)-(7) and (9) (and also implicitly on (1) and (2)). This means that if it were not true that (-x)(-x) = xx, then at least one of these properties would not hold: zero or one would not exist, or you wouldn’t be able to subtract, or results would depend on the order of operands and order of evaluation, or multiplication wouldn’t distribute over addition – or some combination of these. We know that addition and multiplication of rational, real and complex numbers have these properties, and those properties necessarily have as a consequence that (-x)(-x) = xx. Imagine you have a film of a man running forward. If we equate “running forward” with a positive direction and “running backward” with a negative direction, we can set up the following scenarios:
Running Forward (Positive direction): This is straightforward; the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this man moves ahead, symbolizing a positive action or direction (+). Film Played Backwards (Negative Time): If we play the film in reverse, the man appears to be running backwards. This is akin to multiplying a positive by a negative. The direction of motion (positive) is reversed (making it negative), symbolizing a change to the opposite direction or a reversal of what’s expected. Running Backwards on Film (Negative direction): If the man was filmed running backwards (which is already a negative direction), this scenario represents a negative. Film Played Backwards of Man Running Backwards (Negative x Negative): Here’s where it gets interesting. If we play this film in reverse, the man appears to be running forward again. Despite the original action being negative (running backwards), reversing the playback (another negative) corrects the direction back to positive.
Combining two “reversals” or negatives results in a return to the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this original, positive direction. It’s like saying two wrongs (in terms of direction reversal) make a right (return to forward motion). The multiplication of negative numbers can be seen as combining two reversals, changes, or negations, resulting in a positive outcome. If you see a multiplication of a negative number as a rotation by 180° on the number Plane, then you will realise that performing 2 rotations of 180° is the same as not rotating, so (-x)(-x) = xx. You can see that -1 is a primitive square root of unity, so if you square it you get 1, which is the same as not rotating. There are also primitive cube roots of unity, like ω, and -1-ω, which represent a 120° and -120° rotations, and also primitive fourth roots of unity like i and -i, which represent a 90° and -90° rotation, and there are more.
you ve been awarded x(1) points from x(2) jurors how does your score change? +x*x 2.after being penalised by x points from some jurors, x of them retracted their penalty how does your score change now? The word “want” is positive and the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this word “pass” is also positive. Two positives make a positive. However, “do not want” is negative and “fail” is negative. But when you put two negatives together, you end up with a positive in the end. Both the above sentences really mean the same thing even though one has two positives and the other has two negatives. Yeah. You can’t really type -6 in a calculator -6 times to test what -6 * -6 is as a repeated addition, because the least times you can type something is typing it 0 times. Walk forward 6 steps 6 times. You are 36 steps in front of where you started. 66. Now go to the same starting place, turn around, and walk 6 steps backwards 6 times. You end up at the same place. (-6)(-6). If you only turn around OR walk backwards then you go the other direction. That is 6*(-6).
We can also see the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this product (–4) times (6) as a number 6 that we subtract 4 times. So, doing the product of (–4) times 6 is taking away 24, which we write (–4) × 6 = –24. Finally, we can see the product (–4) times (–6) as the number (–6) that we remove 4 times, so it is about removing –24. Removing –24 is adding 24 so (–4) × (–6) = 24. 6* -6 is that -6 is given to you 6 times. You can also say that 6 is TAKEN away from you 6 times. Either way you end up with -36. -6 * -6 is that -6 is taken away from you 6 times. By giving away -6, you end up 6 into the positive. So in the end you have 36. Then we can factor out common multiples, this is arbitrary, but it will help to explain the logic for this example. For those who ask why is it arbitrary? Well there are many different roads to the same destination, so we could express this using different terms, as others have. So if we consider -1 to be a ‘retractration’ and that multiplication is the consecutive operation of the former number applied to the latter, then we are retracting a retraction, which is a correction, to the original. I just want to add one thing: if it’s a fundamental principle, or an axiom you can’t question it. There is no reasoning behind it. It just made sense to do it like this to some people, and the reason behind it really doesn’t matter to understand maths -x*-x is the same as -1x-1x, that’s the same as -1-1xx, and -1*-1=1 because you are facing north, turn around, turn around again, what the hell, you are facing the same direction
Now, by the Danton Heinen Canadian professional ice hockey left wing Vancouver Canucks T-Shirt Additionally,I will love this definition of inverses, (-x) + x = 0. Multiply both sides with (-x) from the right and distribute the (-x), and realise that 0 * (-x) = 0 by the properties of the additive identity 0: When you multiply something by a negative number, you are essentially reversing it. 2 times -2 is -2 – 2. So when you reverse a number already negative, it becomes positive. -2 times -2 is negative two twice reversed. 2 + 2. This was simply decided because it proved to be useful. You can’t justify it any further, since one could also define a mathematics with other such rules… But this rule ensures that positive and negative numbers in operations are “balanced” and have the same frequency: Consider distributive rule: which is -36 And if you have -6 negative six times, the answer HAS to be on the positive side of zero, because (-6) is on the opposite side of zero from 6 So the way I was taught, is to use tiles on a floor, you face north on positive and south on negative, essentially making a number line where north is positive tiles and south is negative tiles. So -6 means face south. The second numbers sign is the direction of your steps, and the times you repeat. So you are at 0 you will take 6 steps, 6 times. In this case since it’s negative it will be backwards. Facing south you take 36 steps backwards. You will have move 36 steps in the north direction. A vector is an entity that has a length and a direction. In one-dimensional space (a line), the direction can have two values (left or right).

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