explain complex number and vectors
Start with real numbers. A number line is often used to represent all real numbers. It has infinite length and somewhere we can mark zero, dividing positive numbers (on the right of zero) and negative numbers (on the left of zero). The line is continuous so no real number can be left out. A line is 1-dimensional.
A complex number has two components: one real, the other imaginary. A complex number can be represented by a plane, and it's 2-dimensional. So what does imaginary mean? The basis of imaginary numbers is the square root of minus one (sqrt(-1)) and traditionally it is given the symbol i. The square root of any negative number can be expressed using i. So, for example, the square root of minus 4 is 2i, because -4=4*-1 and the square root of 4*-1 is 2sqrt(-1)=2i. A complex number, z, can be written a+ib, where a and b are real numbers. But, more importantly perhaps, they can be represented as a point in 2-dimensional space as the point (a,b) plotted on a graph using the familiar x-y coordinate system. So, just like the number line represented all real numbers like an x axis, so all complex numbers can be represented by a plane, an infinite x-y plane.
Now we come to vectors. There is a commonality between complex numbers and vectors. A straight road with cars , houses, people, etc., on it make the road like a number line. Any position on the road can be related to a fixed point on the road we'll call "home". Objects to the right, or eastward, could be in front of home and those to the left, or westward, behind home. The position of an object is the distance from home. This is a 1-dimensional vector field, where all objects have position. If the objects move their speed will have direction, towards the right or towards the left and we can say that a speed to the right is positive and a speed to the left is negative.
Now we introduce another straight road at right angles to the first road. Now the picture is 2-dimensional. The position of an object is defined by two values: position east or west and position north or south. North can be positive and south negative. This is equivalent to the complex plane, which represents all complex numbers. The 2-dimensional plane represents all 2-dimensional vectors, whether it's position or speed. But the word "velocity" is used instead of "speed", because velocity includes direction, but speed is just a number, or magnitude, of the velocity. A vector has an east-west (EW) component (x value) and a north-south (NS) component (y value) and a vector r=xi+yj, where i is called a unit vector in the NS direction and j in the EW direction, so the point (x,y) fixes the positional vector r. Vectors are usually written in bold type, so you won't confuse i with i. The magnitude of a vector is sqrt(x^2+y^2) so it is represented by the hypotenuse of a right-angled triangle whose other sides are x and y. Pythagoras' theorem is used to work out the value. The magnitude of a vector is sometimes written |r| and is called a "scalar" quantity, so it doesn't have a direction or a sign (positive or negative), because the sign is a property of the direction of the vector.
A vector is not limited to a 2-dimensional plane. It can have as many dimensions as necessary. An aeroplane's positional vector and velocity would involve another dimension: height. A submarine's positional vector and velocity would involve depth. Height and depth are perpendicular to the EWNS plane and together form 3-dimensional space. The unit vector for height and depth is k, and height would be positive while depth would be negative, and the letter z is used with x and y so that a point in 3-space is (x, y, z). The magnitude |r| is sqrt(x^2+y^2+z^2).
When working with vectors, addition and subtraction requires adding and subtracting the x, y, z components separately. When adding or subtracting complex numbers, the same applies to the x and y, real and complex, components. Multiplication and division are a special topic beyond the scope of this introductory explanation.
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