Further Pure Mathematics 1

matrix — A matrix is an array of numbers, usually written inside curved brackets.

Matrices are used to represent data or transformations. They consist of rows and columns, with individual entries called elements. Think of a matrix like a spreadsheet or a grid of numbers, where each cell holds a specific value and its position matters.

elements — The entries in the various cells of a matrix are known as elements.

Each number or variable within a matrix is an element. Their position within the matrix (row and column index) is important for matrix operations and defining the matrix's order. If a matrix is a bookshelf, then each book on the shelf is an element.

order — The order of a matrix is stated by the number of rows first, then the number of columns.

For example, a 5 × 5 matrix has five rows and five columns. The order is fundamental as it determines whether matrices can be added, subtracted, or multiplied. The order of a matrix is like the dimensions of a room – length by width.

square matrices — Matrices that have the same number of rows as columns are called square matrices.

Examples include 2 × 2 or 3 × 3 matrices. Square matrices are important for operations like finding determinants, inverses, and representing certain transformations. A square matrix is like a square grid or a chessboard, where the number of rows equals the number of columns.

identity matrix — An identity matrix is a square matrix with 1s on the leading diagonal and zeros elsewhere, usually denoted by I.

When an identity matrix multiplies another matrix, the other matrix remains unchanged, similar to how multiplying a number by 1 leaves it unchanged. Identity matrices must be square. The identity matrix is like the number '1' in scalar multiplication; it leaves the other matrix unchanged when multiplied.

unit matrix — The unit matrix is another name for the identity matrix.

It serves the same purpose as the identity matrix, leaving other matrices unchanged upon multiplication. It is always a square matrix with ones on the main diagonal and zeros elsewhere. Just like a 'unit' of measurement is a standard, the unit matrix is the standard 'multiplicative identity' for matrices.

zero matrix — A zero matrix is a matrix where all elements are zero, usually denoted by O.

When a zero matrix is added to another matrix, the other matrix remains unchanged, similar to how adding 0 to a number leaves it unchanged. Zero matrices can be of any order. The zero matrix is like the number '0' in scalar addition; it leaves the other matrix unchanged when added.

equal matrices — Two matrices are said to be equal if, and only if, they have the same order and each element in one matrix is equal to the corresponding element in the other matrix.

This means that for matrices A and B to be equal, they must have the same number of rows and columns, and aᵢⱼ must equal bᵢⱼ for all i and j. Two matrices being equal is like having two identical jigsaw puzzles; they must have the same number of pieces (order) and each piece must match its corresponding piece in the other puzzle (elements).

non-conformable for addition — Matrices are non-conformable for addition if they are not of the same order.

Matrix addition and subtraction require matrices to have identical dimensions (same number of rows and columns) because elements are added or subtracted in corresponding positions. Trying to add non-conformable matrices is like trying to add apples and oranges; they are different types of items and cannot be combined directly.

scalar number — A scalar number is a single number by which a matrix can be multiplied.

When a matrix is multiplied by a scalar, every element within the matrix is multiplied by that scalar. This operation changes the magnitude of the matrix elements but not its order. Multiplying a matrix by a scalar is like scaling a recipe; if you double the recipe (scalar factor of 2), you double every ingredient (element).

conformable for multiplication — Two matrices are conformable for multiplication if the number of columns in the first matrix equals the number of rows in the second matrix.

For matrices A (order m × n) and B (order n × p), the product AB exists because the 'middle' numbers (n) are the same. The resulting product matrix will have the order m × p. Being conformable for multiplication is like connecting two gears; the number of teeth on the output of the first gear must match the number of teeth on the input of the second gear for them to mesh.

object — The original point, or shape, before a transformation is called the object.

In geometric transformations, the object is the starting figure. Its coordinates or shape are altered by the transformation to produce the image. The object is like the original photograph before you apply any filters or edits.

image — The new point, or shape, after the transformation, is called the image.

The image is the result of applying a transformation to an object. Its position, orientation, or size may differ from the original object. The image is like the edited photograph after you've applied filters or made changes.

transformation — A transformation is a mapping of an object onto its image.

Transformations describe how points or shapes move in a coordinate system. They can be represented by matrices and include reflections, rotations, enlargements, stretches, and shears. A transformation is like a set of instructions for moving or changing a piece on a chessboard; it tells you exactly where the piece ends up.

unit vectors — Vectors that have length or magnitude of 1 are called unit vectors.

In two dimensions, i = (1, 0) and j = (0, 1) are unit vectors along the x-axis and y-axis, respectively. Their images under a transformation matrix form the columns of that matrix. Unit vectors are like the basic building blocks or fundamental directions (north, east) in a coordinate system, each having a standard length of one step.

linear transformations — In a linear transformation, straight lines are mapped to straight lines, and the origin is mapped to itself.

All transformations represented by matrices (reflections, rotations, enlargements, stretches, shears) are examples of linear transformations. This property ensures that the 'straightness' of objects is preserved. A linear transformation is like drawing on a rubber sheet that you can stretch or twist, but you can't tear it or fold it over itself; straight lines remain straight.

shear — A shear is a transformation where points on a fixed line stay the same, and other points move parallel to the fixed line by a distance proportional to their perpendicular distance from the fixed line.

For example, a shear with the x-axis fixed transforms the unit vector j=(0,1) to (k,1), where k is the shear factor. It distorts shapes, turning rectangles into parallelograms. Imagine pushing the top of a stack of cards while keeping the bottom card fixed; the cards slide past each other, creating a slanted stack. This is similar to a shear.

invariant points — Points that map to themselves under a transformation are called invariant points.

For a point (x, y) to be invariant under a transformation represented by matrix M, M(x, y) must equal (x, y). The origin (0,0) is always an invariant point for transformations represented by matrices. An invariant point is like a pivot in a rotation; it's the only point that doesn't move.

invariant line — A line AB is known as an invariant line under a transformation if the image of every point on AB is also on AB.

It is not necessary for each point on the line to map to itself; it can map to another point on the same line. For example, in a reflection, the mirror line is an invariant line of invariant points, while lines perpendicular to the mirror line are invariant lines but not lines of invariant points. An invariant line is like a track on which a train moves; the train (point) moves, but it stays on the track (line).

Sequence — A sequence is an ordered set of objects with an underlying rule.

Sequences can be defined inductively (term-to-term) or deductively (position-to-term). They can be increasing, decreasing, oscillating, or convergent, much like a list of instructions where each step depends on the previous one or its position.

Series — A series is the sum of the terms of a numerical sequence.

Series are often represented using sigma (Σ) notation, indicating the sum of terms from a specified starting limit to an ending limit. The sum can be finite or infinite, similar to adding up scores from each round of a game to get a total.

Inductive definition — An inductive definition describes the relationship between one term and the next in a sequence.

This type of definition requires a starting term (e.g., u1) and a rule to generate subsequent terms (e.g., ur+1 = ur + 3). It is also known as a term-to-term definition, acting like a chain reaction where each event triggers the next.

Deductive definition — A deductive definition describes the relationship between a term and its position in a sequence.

This definition allows you to directly calculate any term in the sequence by knowing its position (r), without needing to know the previous terms. It is also known as a position-to-term definition, like a formula that directly tells you the value of an item based on its slot number.

Increasing sequence — In an increasing sequence, each term is greater than the previous term.

For example, 2, 5, 8, 11, 14 is an increasing sequence because each term is larger than the one before it, implying a positive difference between consecutive terms. This is like climbing a staircase where each step is higher than the last.

Decreasing sequence — In a decreasing sequence, each term is smaller than the previous term.

For example, 14, 11, 8, 5, 2 is a decreasing sequence because each term is smaller than the one before it, implying a negative difference between consecutive terms. This is like walking downhill where each step takes you to a lower elevation.

Oscillating sequence — In an oscillating sequence, the terms lie above and below a middle number.

The terms do not consistently increase or decrease but rather alternate in value, often around a central point, such as 5, -5, 5, -5, ... This is like a pendulum swinging back and forth.

Convergent sequence — The terms of a convergent sequence get closer and closer to a limiting value.

As the number of terms (n) approaches infinity, the terms of the sequence approach a specific finite value, known as the limit. This is similar to a car slowing down as it approaches a stop sign, getting closer to a final position.

Method of differences — The method of differences is a technique to find the sum of a series by expressing each term as the difference of two (or more) terms from a related series, leading to cancellation.

This method is particularly effective when most intermediate terms cancel out, leaving only a few terms at the beginning and end of the series. This is often called a telescoping sum, much like a collapsible telescope where inner sections disappear, leaving only the end pieces.

Telescoping sum — A telescoping sum is a sum where most of the terms cancel out, leaving only the first and last few terms.

This occurs when each term in the series can be written as the difference of two consecutive terms of another sequence, such as f(r) - f(r-1). It's like a stack of cups where only the top and bottom are visible when pushed together.

Proof by mathematical induction — Proof by mathematical induction is a method used to prove that a statement is true for all positive integers.

It involves two main steps: proving the base case (e.g., n=1) and proving the inductive step (if true for n=k, then true for n=k+1). Once these are established, the statement is true for all subsequent integers, much like climbing a ladder where you must get on the bottom rung and then be able to get from any rung to the next.

Polynomial — A polynomial is an expression like 4x³ + x² − 4x − 1, whose terms are all positive integer powers of a variable or multiples of them.

Polynomials are fundamental algebraic expressions, characterized by having only non-negative integer exponents for their variables. They do not contain square roots or reciprocals of variables, simplifying their analysis compared to more complex functions. Think of a polynomial as a 'building block' expression in algebra, made only from whole number powers of a variable, like building with Lego bricks where each brick is a power of x.

Order — The order (or degree) of a polynomial is the highest power of the variable.

The order dictates the maximum number of roots a polynomial can have and influences the general shape of its graph, including the maximum number of turning points. For example, a polynomial of order 3 is called a cubic. The order of a polynomial is like the 'rank' of a military officer; it tells you its highest level of influence and complexity.

Degree — The order (or degree) of a polynomial is the highest power of the variable.

The degree of a polynomial is synonymous with its order and is crucial for determining the number of roots and the maximum number of turning points. It is a fundamental characteristic used in classifying polynomials. The degree of a polynomial is like the 'size' of a house; it gives you an idea of its overall scale and complexity.

Cubic — A polynomial of order 3 is called a cubic.

Cubic polynomials have a highest power of 3, meaning they have three roots (which may be real or complex) and at most two turning points. Their graphs typically exhibit an 'S' shape. A cubic polynomial is like a three-story building; it has a specific structure and a maximum of three 'foundations' (roots).

Roots — The roots of a polynomial equation are the values of the variable where the graph of the polynomial crosses the x-axis (for real roots) or where the equation equals zero.

Roots are the solutions to a polynomial equation, representing the x-intercepts of its graph. A polynomial of order n has n roots, though some may be complex or repeated. Roots are like the 'solutions' to a puzzle; they are the specific values that make the polynomial equation true.

Turning points — The graph of a polynomial function of order n has at most n – 1 turning points.

Turning points are points on the graph where the function changes from increasing to decreasing or vice versa, corresponding to local maxima or minima. The number of turning points is always less than the order of the polynomial. Turning points are like the 'peaks and valleys' on a mountain range; they indicate where the terrain changes direction.

Repeated root — A repeated root occurs when two or more distinct roots coincide to become one.

When a root is repeated, the graph of the polynomial touches the x-axis at that point rather than crossing it. This indicates that the factor corresponding to that root appears multiple times in the polynomial's factorization. A repeated root is like two cars arriving at the same destination at the exact same time; they are distinct entities but occupy the same 'space' (value).

Point of inflection — A point of inflection occurs when two turning points of a curve coincide.

At a point of inflection, the concavity of the curve changes (from concave up to concave down or vice versa), but the gradient does not necessarily change sign. For cubic equations, this can mean only one real root. A point of inflection is like a 'twist' in a road where the curve changes direction without necessarily going uphill or downhill.

Symmetric functions of the roots — Symmetric functions of the roots are functions that remain the same when any two roots are interchanged.

These functions, such as the sum of roots or the sum of products of pairs of roots, can be expressed in terms of the coefficients of the polynomial. This property is crucial for forming new equations or evaluating expressions without explicitly finding the roots. Symmetric functions are like a perfectly balanced seesaw; no matter which two children (roots) swap places, the balance (function value) remains the same.

rational number — A number that can be expressed as p/q where p and q are integers and q ≠ 0.

This definition forms the basis for rational functions, extending the concept of a ratio to algebraic expressions. It highlights the fundamental restriction that the denominator cannot be zero, much like a fraction such as 1/2 or 3/4.

rational function — A function that can be expressed in the form y = f(x)/g(x), where f(x) and g(x) are polynomials, and g(x) ≠ 0.

This is the core concept of the chapter, defining the type of functions being studied. The condition g(x) ≠ 0 is crucial as it leads to vertical asymptotes where g(x) = 0. Think of it as an algebraic fraction, where the numerator and denominator are not just numbers, but expressions involving x.

asymptote — A straight line that a curve approaches tangentially as x and/or y tends to infinity.

Asymptotes are key features of rational function graphs, indicating the behaviour of the function at extreme values of x or y. They are crucial for accurate sketching. Imagine a car driving on a long, straight road that gradually gets closer and closer to a fence without ever touching it; the fence is the asymptote.

vertical asymptote — A vertical line x = a where the function y = f(x)/g(x) is undefined because g(a) = 0.

These occur at x-values where the denominator of a rational function is zero, causing y to tend to positive or negative infinity. They are found by setting the denominator to zero. It's like a 'wall' that the graph gets infinitely close to but can never touch, because division by zero is impossible.

horizontal asymptote — A horizontal line y = c that the curve approaches as x tends to positive or negative infinity.

These describe the end behaviour of the function. They are found by examining the limit of y as x approaches ±∞, often by comparing the degrees of the numerator and denominator polynomials. This is the 'horizon' that the graph approaches as you look far to the left or right, indicating where the y-values settle.

oblique asymptote — A sloping line y = ax + b that a rational function approaches when the degree of the numerator is one greater than that of the denominator.

These asymptotes occur when the function behaves like a linear function for large x. They are found by performing polynomial long division of the numerator by the denominator. Instead of approaching a flat horizon, the graph approaches a sloping 'ramp' as x gets very large.

stationary point — A point on a curve where the gradient is zero, i.e., dy/dx = 0.

These points correspond to local maxima, local minima, or points of inflection. Finding them is crucial for accurately sketching the shape of the curve and determining its range. Imagine a ball rolling on a hill; a stationary point is where the ball momentarily stops at the very top (maximum) or bottom (minimum) of a dip.

range — The set of all possible y-values that a function can take.

Understanding the range is essential for fully characterising a function's behaviour. It can be determined from the sketch, by using differentiation to find turning points, or by using the discriminant method. If the domain is all the ingredients you can put into a recipe, the range is all the possible dishes you can make.

even function — A function f(x) for which f(x) = f(-x), meaning its graph is symmetrical about the y-axis.

Recognising even functions simplifies sketching as only one side of the y-axis needs to be drawn, and the other can be reflected. This saves time and reduces potential errors. It's like folding a piece of paper along the y-axis; the two halves of the graph would perfectly overlap.

odd function — A function f(x) for which f(x) = -f(-x), meaning its graph has rotational symmetry of order 2 about the origin.

Like even functions, identifying odd functions aids in sketching by allowing reflection about the origin. This property is less common but equally useful when present. If you rotate the graph 180 degrees around the origin, it looks exactly the same.

modulus — The absolute numerical value of a number or expression, always taking a positive value.

The modulus function, denoted |f(x)|, transforms any negative y-values of f(x) into their positive counterparts, effectively reflecting parts of the graph below the x-axis. It's like a 'positive-only' filter; whatever number you put in, you only get its positive version out.

polar coordinates — The numbers (r, θ) that describe the position of a point P by its distance r from a fixed point O (the pole) and the angle θ between OP and an initial line.

Polar coordinates offer an alternative to Cartesian coordinates, particularly useful for curves with rotational symmetry. The distance r is always non-negative, and the angle θ is typically measured in radians anticlockwise from the initial line. Imagine a radar screen: 'r' is how far away a target is from the center, and 'θ' is its bearing (direction) from a fixed reference line.

pole — A fixed point O from which the distance r to a point P is measured in polar coordinates.

The pole serves as the origin in the polar coordinate system, analogous to the origin (0,0) in Cartesian coordinates. At the pole, r=0 and θ is undefined. The pole is like the bullseye of a dartboard; all distances are measured outwards from it.

initial line — A line drawn horizontally to the right from the pole, used as the reference for measuring the angle θ in polar coordinates.

The initial line is the angular reference (θ=0) in the polar system, similar to the positive x-axis in Cartesian coordinates. Angles are typically measured anticlockwise from this line. Think of the initial line as the starting point on a protractor, from which you measure angles.

principal polar coordinates — A unique set of polar coordinates (r, θ) for a point where r > 0 and −π < θ < π.

These coordinates provide a unique representation for each point in the plane, resolving the issue of infinite possible (r, θ) pairs for a single point. This convention is similar to the modulus-argument form for complex numbers. If polar coordinates are like having many nicknames, principal polar coordinates are like having one official name that everyone uses.

r — The distance of a point P from the pole O in polar coordinates.

r is the radial distance and is always considered non-negative in the standard convention. It forms the hypotenuse of a right-angled triangle when converting to Cartesian coordinates. r is like the radius of a circle that passes through the point P, centered at the pole.

θ — The angle between the line segment OP and the initial line in polar coordinates.

θ is typically measured in radians, in an anticlockwise direction from the initial line. It determines the angular position of the point P. θ is like the hand on a clock, but instead of hours, it measures the angle from a fixed starting point.

x — The Cartesian coordinate representing the horizontal distance from the y-axis.

In the conversion between polar and Cartesian systems, x is related to r and θ by x = r cosθ. It is the horizontal component of the point's position. If you walk 'r' steps at an angle 'θ', 'x' is how far you've moved horizontally from your starting point.

y — The Cartesian coordinate representing the vertical distance from the x-axis.

In the conversion between polar and Cartesian systems, y is related to r and θ by y = r sinθ. It is the vertical component of the point's position. If you walk 'r' steps at an angle 'θ', 'y' is how far you've moved vertically from your starting point.

determinant — The quantity (ad − bc) is the area scale factor associated with the transformation matrix.

For a 2x2 matrix, the determinant indicates how much the area of a shape is scaled by the transformation. A negative determinant signifies a reversal in the order of vertices, like flipping a rubber sheet over. For a 3x3 matrix, it represents the volume scale factor.

singular — If the determinant of a matrix is zero then the inverse matrix does not exist and the matrix is said to be singular.

A singular matrix maps all points on the plane to a straight line (in 2D) or to a plane (in 3D), meaning an infinite number of points map to the same image point. This makes it impossible to uniquely reverse the transformation, hence no inverse exists. It's like trying to reverse a process where many different starting points lead to the exact same outcome.

non-singular — If det M ≠ 0 the matrix is said to be non-singular.

A non-singular matrix has a non-zero determinant, which means its inverse exists. This type of matrix represents a transformation that can be uniquely reversed, as it does not collapse dimensions. It's like a reversible operation, such as stretching a rubber band; you can always stretch it back to its original state.

identity matrix — The matrix \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} is known as the 2 × 2 identity matrix, which maps every point to itself.

Identity matrices, denoted by I, behave like the number 1 in real number multiplication; when multiplied by any matrix M, the product is M. This matrix represents a transformation that leaves all points unchanged, like doing nothing at all to a shape.

inverse — If the product of two square matrices, M and N, is the identity matrix I, then N is the inverse of M, written as N = M−1.

The inverse matrix 'undoes' the transformation represented by the original matrix. If a transformation A maps P to Q, then A−1 maps Q back to P. An inverse only exists for non-singular matrices. It's like a set of instructions to move an object back to its original position after it has been moved.

minor — The 2 × 2 determinant obtained by deleting the row and column containing an element is called the minor of that element.

For a 3x3 matrix, each element has a corresponding minor. These minors are crucial intermediate steps in calculating the determinant and the cofactors, which are then used to find the inverse matrix. Think of a minor as a 'sub-determinant' or a smaller puzzle piece within the larger matrix puzzle.

cofactor — A minor, together with its correct sign, is known as a cofactor and is denoted by the corresponding capital letter.

The sign of a cofactor depends on its position in the matrix, following an alternating pattern (+ - + / - + - / + - +). Cofactors are used in the expansion of the determinant and in forming the adjugate matrix for finding the inverse. If a minor is the raw value, the cofactor is that value with a 'directional' sign attached.

adjugate — The matrix formed by replacing each element of M by its cofactor and then transposing the matrix (i.e. changing rows into columns and columns into rows) is known as the adjugate or adjoint of M, denoted adj M.

The adjugate matrix is a key component in the formula for finding the inverse of a 3x3 matrix. It is derived from the matrix of cofactors, which is then transposed. If the cofactors are individual pieces of a puzzle, the adjugate matrix is like arranging those pieces into a new, specific pattern (transposed) that, when combined with the determinant, reveals the inverse.

magnitude — The length or size of a vector.

Magnitude is a scalar quantity that describes the extent of a vector, such as the speed of an object or the strength of a force. If a vector is an arrow, its magnitude is the length of the arrow.

direction — The orientation of a vector in space.

Direction specifies where a vector is pointing, often given by angles relative to coordinate axes or as a unit vector. If a vector is an arrow, its direction is where the arrow is pointing.

position vector — A vector that represents the position of a point in space relative to an origin.

A position vector starts at the origin and ends at the point, uniquely defining the point's location. Think of a position vector as the GPS coordinates for a specific location, but expressed as a vector from a central starting point.

scalar product — A product of two vectors that results in a scalar quantity, calculated as a · b = |a||b|cosθ.

The scalar product (also known as the dot product) measures the extent to which two vectors point in the same direction and is useful for finding the angle between vectors or checking for perpendicularity. Imagine two people pulling ropes; the scalar product tells you how much of their effort is aligned, contributing to movement in the same direction.

normal to the plane — A line or vector that is perpendicular to every line in a given plane.

The normal vector defines the orientation of the plane in space; its direction is unique (up to a scalar multiple) for any given plane. If a plane is a flat table, the normal vector is like a pencil standing perfectly upright on that table, perpendicular to its surface.

vector product — A product of two vectors that results in a vector perpendicular to both original vectors, calculated as a × b = |a||b|sinθ n̂.

The vector product (also known as the cross product) produces a vector whose magnitude is the area of the parallelogram formed by the two vectors, and whose direction is given by the right-hand rule, perpendicular to the plane containing the two vectors. If you have two vectors representing forces, their vector product can represent a torque, which is a rotational force perpendicular to both original forces.

anti-commutative property — A property of the vector product where a × b = − b × a.

This means that reversing the order of the vectors in a vector product reverses the direction of the resulting vector, while its magnitude remains the same. If turning a screw clockwise drives it in, turning it anti-clockwise drives it out; the action is the same, but the direction is opposite.

sheaf of planes — An arrangement where several planes share a common line of intersection.

When two planes intersect, they form a line; any other plane containing this same line is part of the sheaf, meaning all planes in the sheaf pass through that common line. Think of the pages of an open book; each page is a plane, and the spine of the book is the common line of intersection for all those 'planes'.

skew lines — Two lines in three dimensions that do not intersect and are not parallel.

Skew lines exist in different planes and never meet, unlike parallel lines (which are in the same plane) or intersecting lines (which meet at a point). Imagine two aeroplanes flying at different altitudes and on different headings; they might pass close to each other but never collide if their paths are skew.