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Argand diagrams are, like graphs, a visual representation. Instead of representing an equation though, Argand diagrams represent numbers.
Any number can be represented on an Argand diagram, be it real, imaginary or complex. Mostly, though, it is used to show complex numbers.
As you probably know, complex numbers are made up of a combination of real and imaginary numbers. For example 3+2i.
An Argand diagram has two axes, like a regular cartesian graph. However, in this case the horizontal axis represents real number, just like an ordinary number line you learnt about in primary school. The vertical axis represents imaginary rumbers. The axes cross at zero, again just like in a cartesian graph.
To plot 3+2i on an Argand diagram, you plot the point where the value on the real axis reads 3 and the value on the imaginary axis reads 2i. Then, extend a line from 0 to the point you just plotted. That line is the visual representation of the number 3+2i.
Some other properties are represented by the line on the Argand diagram. The length of the line represents the modulus of the number: √(32+22) = √13
The line also forms an angle with the positive side of the real axis. This angle, measured in radians, is known as the argument, and is important when representing complex numbers in polar form. For many aspects of dealing with complex numbers, this form can be a lot more useful.
So you can see, just from one line on an Argand diagram, you can learn a lot of the information you need to manipulate a complex number.see more
Practical applications of calculus occur more often than you would think. The most common example of them being applied to real world problems is through the relationship between distance and time. The distance an object will travel in a certain time is it's velocity. If we differentiate an equation for distance travelled with respects to time, it will give an equation for velocity.
However this is only useful if the objects velocity is constant; it could be speeding up (acccelerating). The acceleration of a body is it's change in speed over time. So, if we differentiate an equation for velocity with respects to time, it will give an equation for acceleration.
You can differentiate again to give the rate of change of acceleration, but this is a less useful figure to be given.
Conversely, if you have an equation for an object's acceleration, you can integrate it (the opposite of differentiating it) with respects to time to give an equation for it's velocity at a certain time.
Integrating again will give the distance it has travelled after a certain time.
Calculus can be used in this way in loads more equations including force-energy-power, and circumference-area in a circle.
All of these values can be plotted on graphs, and different features of a particular graph (such as the gradient of the line and the area under it) will give you these values. Remember, a graph is just a visual representation of an equation, so that's where they come into it.