Campbell McGregor, Glasgow, UK Whilst being whimsical for an eccentric mathematician, imaginary numbers can be very useful for solving engineering problems. On example is if you have a pendulum swinging, it starts to slow down and eventually stop.
If you want to work out the motion of the pendulum over a certain time ie derive a formula then the best way to do it is to use complex numbers. David Vickery, Croydon, UK If you're talking about things like the square root of minus one, then they have all sorts of applications.
For example, if I recall my physics imprecisely the two-dimensional number matrix formed by real numbers and multiples of "i" i. Richard, London, UK Imaginary, or complex, numbers aren't much use when adding up your shopping bill or working out your tax, on second thoughts As an example, you probably wouldn't have the weather forecast if it wasn't for imaginary numbers.
Although forecast models don't use complex numbers themselves though you may think they do , the mathematical theories on which the models are based rely on them. Raymond Lashley, Reading, UK They find ample application, along with all those sines, cosines and tangents and the rest of your high school math, in many areas of engineering such as electronics and electrical engineering. Rather than wanting to actually evaluate the square root of minus one it is handy to have something that when squared is minus one.
It's best illustrated with a simple circle and sine wave. Complex numbers come into place whenever one force gets divided into two or more components due to inclination or whatever other reason.
There are more that one way an object can be inclined and thus more than one way these forces get divided into two. The i, j and k planes are a resultant of this. Mathematician says:. Mathemusician says:. January 6, at am. John Gabriel says:. August 7, at pm. The reply by Mathematician is anything but clear. Frankly, it is nonsense. Matt says:. August 15, at pm. Ron says:. May 20, at am. June 25, at pm. The Physicist says:. Andy says:. February 11, at am. Jeruel Camat says:.
March 11, at pm. Xerenarcy says:. March 21, at am. Stan says:. October 14, at am. January 17, at pm. April 27, at am. Rajesh Swarnkar says:.
June 24, at am. Ben Ford says:. The introduction of this one new non-real number — i , the imaginary unit — launched an entirely new mathematical world to explore. It is a strange world, where squares can be negative, but one whose structure is very similar to the real numbers we are so familiar with.
And this extension to the real numbers was just the beginning. The quaternions are structured like the complex numbers, but with additional square roots of —1, which Hamilton called j and k. For instance, will the system be closed under multiplication? Will we be able to divide? Hamilton himself struggled to understand this product, and when the moment of inspiration finally came, he carved his insight into the stone of the bridge he was crossing:. People from all over world still visit Broome Bridge in Dublin to share in this moment of mathematical discovery.
The other products can be derived in a similar way, and so we get a multiplication table of imaginary units that looks like this:. Notice this means that, unlike with the real and complex numbers, multiplication of quaternions is not commutative. Multiplying two quaternions in different orders may produce different results! To get the kind of structure we want in the quaternions, we have to abandon the commutativity of multiplication.
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. There is also an interesting property of i.
When you multiply it, it cycles through four different values. This makes exponents of i easy to figure out. This cycle will continue through the exponents, also known as the imaginary numbers chart. Knowledge of the exponential qualities of imaginary numbers is useful in the multiplication and division of imaginary numbers.
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