Complex Numbers Year 12 QLD Maths C

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1 Complex Numbers Representations of Complex Numbers We can easily visualise most natural numbers and effectively use them to describe quantities in day to day life, for example you could describe a group of cows by saying there is 8 of them. Fractions are not as simple and neither are negative numbers, although they both have their uses. Mathematics has more abstract ways of representing numbers as surds and other irrational numbers that can t be used effectively for day to day tasks, you can t describe a group of cows by saying there is 8 of them. Another such abstract of representation of quantity is Complex Numbers. Standard Form It s well known that you can t take the square root of a negative number But through algebraic manipulation you can just about always get down to a multiple of 1 In calculations involving complex numbers, the letter i is used to represent the square root of -1 and letters like z and w are commonly used to represent complex numbers The standard form of a complex number is z = a + bi Where z is the complex number The letters a and b are real numbers Complex numbers have real and imaginary components The Real component is represented by a and could be placed anywhere on a number line The imaginary component is represented by bi and does not fit on the ordinary number line The short hand for the real and imaginary components of the complex number z is Re(z) and Im(z) respectively Similarly, for an imaginary number called w the components could be represented as Re(w) and Im(w) For the above complex number Re (z) = a Im (z) = bi If b is equal to zero, the complex number has no imaginary component and is essentially a real number Because of this all real numbers like 0.1, 10 and 100 can be considered as complex numbers where the coefficient of i is zero

2 Argand Diagrams Before it was mentioned that the Real component of a complex number can be placed on a number line but the imaginary component can t. Argand diagrams work by crossing the real number line against the imaginary number line to create a graph. They are a common way of representing complex numbers. On an argand diagram, complex numbers may be represented as a vector, a line or a point The axes are called the real and imaginary axes. Both have real number scales and meet at the Origin (O) where the values for both are 0. The angle θ is measured from the positive real axis to the point in the shortest way possible ( π < θ π) Angles below the real axes are taken as negatives. The distance between the origin and the point is called the magnitude or modulus and is represented as r Polar Form The standard form of a complex number is the sum of the real and imaginary components, but the polar form is something quite different. z = rcis(θ) r and θ the quantities displayed on the graph cis is mathematical shorthand that is explained below If not given directly r (also written as z ) can be determined using Pythagoras rule And θ can be determined by basic trigonometry r = z = a 2 + b 2 θ = tan 1 ( b a ) The quantities of r and θ are called the modulus and argument respectively For the complex number z, they are represented by the abbreviations mod(z) and arg(z) Also through basic trigonometry the values of a and b can be determined from the modulus and argument.

3 Matrices Review of Matrices A matrix is unlike surds, complex numbers and fractions in that it is not a representation of a number. Instead it s kind of like a container for numbers that is very abstract but somehow works to allow effective, efficient and unique calculations Think of a matrix as a table of numbers without the borders The proper word for it is array The numbers within a matrix are called its elements The dimensions of a matrix are called its order All notation with matrices has the number of rows, followed by the number of columns The matrix on the right as an order of 4 3 Individual elements within a matrix are described by their placement; how many rows down they are and how many rows across they are where the top left is the first row/column A lower-case letter if used to denote an element of the same upper case letter matrix, where i and j refer to the number of the row and column the element is in, respectively Matrices are always represented as capital letters A = [ ] a ij E.g.: a 23 = 3 a 42 = 9 Operations with Matrices Addition and Subtraction To add or subtract matrices, they must be of the same order The corresponding elements in each matrix are added/subtracted to form one matrix of the same order a b f + e b + f [ ] + [e ] = [a c d g h c + g d + h ] a b f e b f [ ] [e ] = [a c d g h c g d h ] Matrices of different order cannot be added or subtracted E.g.: [ a b e f c d ] + [ g h] cannot be done i j

4 Multiplication The most important thing about matrix multiplication is to remember that it is not commutative, A B B A For multiplication to be possible, the first matrix must have the same number of columns as the second matrix has rows Matrices that can be multiplied are described as conformable The process for multiplying matrices is hard to describe, each row in the first matrix is multiplied by each column in the second matrix works better if you can see it with a few examples a b f + bg af + bh [ ] [e ] = [ae c d g h ce + dg cd + fh ] a [ c e b d] [ g h ag + bi ah + bj i j ] = [ cg + di ch + dj] f eg + fi eh + fj x a b c [ d e f ] [ ax + by + cz y] = [ dx + ey + fz ] z You may have noticed a pattern that dictates the order of the resulting matrix, that is, it will have the same number of rows as the first matrix and the same number of columns as the second This will be helpful to remember when reintroducing yourself to multiplying matrices There is one matrix, the identity matrix (I), that makes for commutable multiplication It is the matrix equivalent of the number 1, in that where a 1 = a = 1 a, A I = A = I A The identity matrix is always square, and has 1 s down the leading diagonal and 0 s everywhere else

5 Vectors Review of Vectors Vectors are used generally for mechanics/kinematics kind of stuff Represent magnitude and direction Direction is pretty straightforward, but the magnitude can be things like force, velocity, acceleration Vectors are denoted in several ways, because I m typing this they ll be capital letters with an arrow above, but other ways include bolding or putting a tilde (~) underneath The unit vectors are the backbone of everything you ll need to do with vectors They are all 1 unit long The i, j, k vectors extend in the x, y, z directions respectively Every other vector is essentially some combinations of multiples of the unit vectors, e.g. the vector OA = 2i + 3j k, represents a vector going from the origin, 2 units in the x direction, 3 units in the z direction, and -1 units in the z direction Depending on which of the i, j, k vectors they consist of, a vector can be in 1, 2 or 3-dimensional space The form of vectors above is called component form, that is when the vector is written as the combination of multiples of the unit vectors Pretty much the same as complex numbers, in that it can go in polar form, but only really when there s 2 dimensions. It s the exact same deal, but without the cis abbreviation When two vectors are added, the vector produced is called the resultant vector Something they don t really teach this, but I think makes it a lot easier to do calculations is matrix form Basically, just a column matrix (which for this exact reason is often called a vector) Each element represents a component from component form a E.g.: v = ai + bj + ck would be represented by the matrix [ b] c If you know you need to do a lot of basic operations like resultant vectors and multiplying by scalar, it will be worth your time to use matrix form

6 Conics Introduction to Conics Conics is probably the biggest unit you ll do in year 12 Maths C, and it s one of the only units that isn t explored in year 11. It deals with the geometry of cross sections (called conic sections) taken from cutting two cones joined at the apex at different angles. Other than knowing that, there isn t really anything else to do with cones. All conics on the Cartesian plane follow the general second degree equation 0 = Ax 2 + Bxy + Cy 2 + Dx + Ey + F All conic sections can be classified into 5 categories based on their eccentricity Each category has different conditions for the general second degree equation Conic Section Category and Conditions Eccentricity Circle A = C > 0, B = 0 0 = Ax 2 + Ay 2 + Dx + Ey + F Ellipse A and C > 0, B = 0 0 = Ax 2 + Cy 2 + Dx + Ey + F Parabola (Vertical) B = C = 0, or (Horizontal) A = B = 0 0 = Ax 2 + Dx + Ey + F 0 = Cy 2 + Dx + Ey + F hyperbola B = 0, and A > 0, C < 0 or A < 0,C > 0 0 = Ax 2 Cy 2 + Dx + Ey + F 0 = Cy 2 Ax 2 + Dx + Ey + F Straight Line A = B = C = 0, 0 = Dx + Ey + F

7 With the exception of the straight line, all conic sections are curved lines In conics, the line(s) that makes up a section is called a locus It s defined as the path traced out by a moving point P To make conic sections certain conditions are put of the movement of the point P, like it has to be the same distance from one set point as it is from a set line, or the combined distance from two set points must be constant For these reasons, it doesn t hurt to remember these formulas Distance from a point Distance from a line d = (x 2 x) 2 + (y 2 y) 2 d = Ax 1 + By 1 + C A 2 + B 2 Parametric and Polar Equations Most of what you ve done in year 11 and 12 have used position coordinates. Through vectors and complex numbers you will have been made familiar with polar coordinates however not on the Cartesian plane. The Cartesian plane is what you ve been using since year 9, with an x axis and y axis. Position coordinates give the location of a point by showing how far it is from the origin along the x and y-axis, whereas polar coordinates can show the same position in terms of radial distance from the origin (pole) and angle from the positive x-axis (polar axis). Position coordinates take form of P(x, y) Polar coordinates take the form of P(r, θ) In both coordinate systems, the second coordinate is generally given as a function of the first E.g.: y = f (x) and r = f (θ) Position coordinates are really good for straight lines and that kind of stuff, but you can t model a relation like a circle using a y = f (x) system, because it can only produce 1 y coordinate for every x coordinate Similarly, Polar coordinates are pretty good for curves and arcs, but you can t model a spiral using a r = f (θ) because there s more than 1 r value for every θ

8 Integral Calculus Introduction There are two main types of calculus, integral and differential. If you have a function f (x), the derivative d dx (f (x)) will tell you the gradient of the function at any value for x In contrast the integral f (x) dx will tell you the area between the function and the x-axis from the origin to any value of x Integration and differentiation are opposite processes, just as you can undo multiplication with division of the same number, you can undo differentiation with integration and vice versa This is the first fundamental theorem of calculus Largely the reason integration is more often called antidifferentiation In the above graph, the red line represents f (x), the blue line d dx (f (x)) and the green area f (x) dx The area below the x-axis counts as negative area The integral can also take the form of a function that can be plotted just like f (x) and d dx (f (x)), as shown by the darker green line is the symbol for the integral The function that follows it is called the integrand It is always followed by a dx or whatever the appropriate variable is, to show that it is to be integrated with respect to x The derivative divides by dx, the integral multiplies by dx For the function f (x), it s derivative may also be written as f (x) And its integral as F (x)

9 Dynamics Introduction Dynamics and kinematics deal with modelling the motion of objects. For Year 12 Maths C, an approach that would be described as Newtonian in physics, is taken to dynamics. All objects, whether it be a plane or a football are treated as individual particles that only take up a set point in 1,2 or 3-dimensional space and move accordingly. This approach neglects the shapes and weight distribution of the objects it s modelling and is for that reason flawed Motion in a straight line is called rectilinear motion, think of it as motion in 1 dimensional space The difference between a scalar and a vector is significant in dynamics - A Scalar is just a number that represents the magnitude of something, e.g.: 10 km/h - A Vector is a scalar with a direction, e.g.: 10 km/h west It might not seem like much but it really is a big deal, because the motion of any object can only be described with respect to the motion of another object when you throw a ball, the ball only moves because you and your environment are not moving with it Because of this the direction component of a vector makes quite a big difference With rectilinear motion, because there is only one dimension, the direction of a vector can be described simply with a + or sign Recall that when typed vectors are represented by either a bold letter (a) or an arrow (OA ), and when written they are underlined with a tilde (~) In dynamics, there are four main vectors; position, displacement, velocity, and acceleration The position vector (r) represents the distance and direction from the origin The displacement vector (x) is defined as the change in the position vector x = r (t+ t) r (t) What this equation means is that displacement is equal to the new position minus the old position By performing vector subtraction, the vector x is created representing displacement If r (t=0) is at origin, as it is in most dynamics situations, then you can see on the graph how the displacement vector x would be identical to vector r (t+ t) Otherwise if you were to map out the all the points of the position vector, to illustrate the path, the displacement vector would look nothing like it

10 Structures and Patterns Sequences and series If you have a long set of numbers like the one below, mathematical sequences and series are used to summarise them. This is a big deal for IT and that kind of stuff because the use of sequences and series can really save a lot of data 3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63 If you needed to communicate this set of numbers to someone, reading out the numbers in the set wouldn t be effective, you d say something faster like start at three and go up by four. That s basically how sequences and series work, they summarise large sets of data using rules. There are three types of sequences and series you ll need to know for year 12 A series refers to the actual set of numbers, whereas the sequence refers to the rule that defines them Each number in a series is called a term (T) Sequences are usually given as a function that equals any term T n Where n is called the term number The terms in a series are numbered, in the above series, 7 is the second term (T 2 ) In most cases the first term (a) of the series is also need to make a rule, in the above series, 3 is the first term Arithmetic An arithmetic sequence is where there is a constant difference (d) between terms Basically, it means that you keep adding the same number to every term In the above example the constant difference is four The general rule for an arithmetic sequence is T n = a + (n 1)d This can be used to find any term in the series To mathematically prove that a series is arithmetic, you must show that the constant difference exists between two separate pairs of consecutive terms d = T n T n+1 Basically just choose two different numbers for n and show that d = d The sum of all the numbers in a series up to any term S n can also be computed s n = n [2a + (n 1)d] 2