![]() ![]() ![]() So, once the natural numbers have been defined, one proceeds to define what a whole number is. That’s because in maths EVERYTHING is a set. Now, one of the axioms says that there exists at least one infinite set that is inductive (this property is needed to DEFINE what a natural number is). Then, one starts deriving logical consequences – theorems. In the set theory, one first postulates the existence of some objects that are called sets and lays out the statements/axioms that describe some properties of the objects without a proof. The easiest way to tell people that – * – = + is that this follows as a logical consequence of the axioms of the set theory. If either operand is negative then the result is negative, if both are negative then the result is positive. However division for negative numbers uses exactly the same rules as for multiplication of negative numbers. Instead division by permutation uses an indirect iterative process using multiple addition and manipulation of the denominator to create a temporary token value which is then compared and adjusted repeatedly until it adequately fits to the value of the nominator. – Direct division is generally actually incomputable using simple operators. Although division is the inverse of multiplication it is not quite the same procedure. A negative addition is simply a subtraction so the permutation here becomes a ‘multiple subtraction’. If you are multiplying by zero you end up with zero copies of the original number which of course adds up to zero.Įxtending this a negative multiplication becomes a negative multiple addition. Multiplication is really just a specialized form of addition and so ‘multiplication’ is really just ‘multiple addition’. This is the kind of area where looking at the theory of permutation can be useful. Specifically, you’ll discover why arithmetic’s shackles are usually left unshaken. ![]() It’s well worth your time to shake off the shackles of mundanity and conformity, so that you can forge into a world of new discoveries. Using the wrong rules is a good, practical, and genuinely useful training in what not to do. Using a different rule means asking a lot of hard questions, like: What is negativeness? Which rules of arithmetic are worth keeping? What is the sound of negative two hands unclapping? But when you look at lots of examples and the number system overall, you find that the “ ” rule is kinda hard to avoid. On a case-by-case basis, it’s not obvious that a negative times a negative should be positive. In other words, we need to use “ ” in order for arithmetic to work.Īnd if you begrudgingly allow the “ ” rule, but refuse to accept the “ ” rule, then consider this: and. The discerning eye will note that 5≠25, so. īut if you insist on using the rule “ “, then you’ll find the distributive property doesn’t work. Losing the distributive property basically means you need to go home and start designing a new (and worse) kind of math from scratch.įor positive numbers there’s no issue, because (practically) everyone is fine with the “ rule”. In fact, this property is literally the thing that defines the relationship between addition and multiplication! For example,īecause “3” is defined as “3=1+1+1”. In particular, the distributive property, which says that, is one of the backbone rules upon which all of arithmetic is built. But does it work with the rules of arithmetic? So, “, , and ” is a clean, reasonable way to define multiplication. In some sense, the rules for signs are set up so that multiplication tables like this follow a nice, simple pattern. ![]() If the “negative times a negative” quadrant on the lower left were all negative instead of positive (e.g., “ “), then the rows and columns that go through it will suddenly have to switch patterns (e.g., “increasing by 3’s” to “decreasing by 3’s”) when they pass zero. While you’re noticing things, notice that the pattern is always the same for each row and column, even when the sign changes. Notice that the “-3 row” always increases by -3 (which is to say, decreases by 3) every step to the right. Notice that the “2 row” always increases by 2 every step to the right (…, -2, 0, 2, 4, …). The times table for numbers between -3 and 3. ![]()
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