This is a real loss: Commutativity is a kind of algebraic symmetry, and symmetry is always a useful property in mathematical structures. But with these relationships in place, we gain a system where we can add, subtract, multiply and divide much as we did with complex numbers.
To add and subtract quaternions, we collect like terms as before. To multiply we still use the distributive property: It just requires a little more distributing. And to divide quaternions, we still use the idea of the conjugate to find the reciprocal, because just as with complex numbers, the product of any quaternion with its conjugate is a real number. Thus, the quaternions are an extension of the complex numbers where we can add, subtract, multiply and divide.
And like the complex numbers, the quaternions are surprisingly useful: They can be used to model the rotation of three-dimensional space, which makes them invaluable in rendering digital landscapes and spherical video, and in positioning and orienting objects like spaceships and cellphones in our three-dimensional world.
And just as with the quaternions, we need some special rules to govern how to multiply all the imaginary units.
As in the representation for the quaternions, multiplying along the direction of the arrow gives a positive product, and against the arrow gives a negative one. Like the quaternions, octonion multiplication is not commutative. But extending our idea of number out to the octonions costs us the associativity of multiplication as well.
For example, using the diagram above, we can see that. So now we have a number system with non-commutatitve, non-associative multiplication and seven square roots of When would anyone ever use that? Well, some physicists believe that the octonions may hold the key to describing how the strong, weak and electromagnetic forces act on quarks, leptons and their anti-particles. If true, this could help resolve one of the great mysteries in modern physics.
By repeatedly extending the real numbers to create larger systems — the complex numbers, the quaternions, the octonions — in which we can add, subtract, multiply and divide, we lose a little familiarity with each step. Along the way, we may also lose touch with what we think of as real. But what we gain are new ways of thinking about the world. And we can always find a use for that. Under what conditions on a and b would this be equal to i? Can you find the other two cube roots of —1?
This book was a great event in mathematics. In fact, it was the first major achievement in algebra in years, after the Babylonians showed how to solve quadratic equations. Cardano also dealt with quadratics in his book. Unable to display preview. Download preview PDF. Skip to main content. There's no answer. You can't have negative five apples, right? But think of it this way. You could owe me five apples, or five dollars.
Once people started doing accounting and bookkeeping, we needed that concept. Another way to look at negative numbers — and this will come in handy later — is to think of walking around in a city neighborhood, Moore says. If you make a wrong turn and in the opposite direction from our destination — say, five blocks south, when you should have gone north — you could think of it as walking five negative blocks to the north.
Imaginary numbers and complex numbers — that is, numbers that include an imaginary component — are another example of this sort of creative thinking. As Moore explains it: "If I ask you, what is the square root of nine, that's easy, right? The answer is three — though it also could be negative three," since multiplying two negatives results in a positive. But what is the square root of negative one? Is there a number, when multiplied by itself, that gives you in negative one?
But Renaissance mathematicians came up with a clever way around that problem. Let's give it a name. Once they came up with the concept of an imaginary number, mathematicians discovered that they could do some really cool stuff with it. Remember that multiplying a positive by a negative number equals a negative, but multiplying two negatives by one another equals a positive. But what happens when you start multiplying i times seven, and then times i again?
Because i times i is negative one, the answer is negative seven. But if you multiply seven times i times i times i times i , suddenly you get positive seven. Now think about that.
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