You can also understand their instability and progress to turbulence. All of the above are relevant in the real world, as they give insight into how to pump oil in oilrigs, how earthquakes shake buildings and how electronic devices such as transistors and microchips work on a quantum level increasingly important as the devices shrink. Gareth Owen, Crewe UK Ask any phisical scientist or engineer mechanical, civil or electrical how they would get on without using the square root of minus one. They will tell you most of our technology depends on it.
For example, without using imaginary numbers to calculate various circuit theories, you would not be reading this on a computer. G Baker, Ockendon, UK Yes, electrical engineers use them as they are a mathematical representation of alternating current.
They use 'j' to represent the square root of -1 unlike mathematicians who use 'i' since in electrical engineering 'i' represents "current". 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. In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.
AC Alternating Current Electricity changes between positive and negative in a sine wave. When we combine two AC currents they may not match properly, and it can be very hard to figure out the new current. The beautiful Mandelbrot Set part of it is pictured here is based on Complex Numbers. The Quadratic Equation , which has many uses, can give results that include imaginary numbers. The Unit Imaginary Number, i , has an interesting property. 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. You took an imaginary number, plugged it into an equation multiple times, and ended up with an actual number that you commonly use in the real world. It wasn't until few hundred years later, in the early s, that mathematicians discovered another way of understanding imaginary numbers, by thinking of them as points on a plane, explains Mark Levi.
When we think of numbers as points on a line, and then add a second dimension, "the points on that plane are the imaginary numbers," he says. Envision a number line. That's why it's positive," Levi explains. But you can't put the square root of negative one anywhere on the X axis. It just doesn't work. However, if you create a Y axis that's perpendicular to the X, you now have a place to put it. And while imaginary numbers seem like just a bunch of mathematical razzle-dazzle, they're actually very useful for certain important calculations in the modern technological world, such as calculating the flow of air over an airplane wing , or figuring out the drain in energy from resistance combined with oscillation in an electrical system.
And the fictional Robert Langdon wasn't pulling our legs when he mentioned that they're also used in cryptography. Complex numbers with imaginary components also are useful in theoretical physics, explains Rolando Somma , a physicist who works in quantum computing algorithms at Los Alamos National Laboratory.
Thus, as in math, complex calculus in physics is an extremely useful tool for simplifying calculations.
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