Engineering Maths 1 Flashcards
Constant
A value or number that never changes in an equation.
Exponent
A small number written slightly above and to the right of a variable or number. It’s used to show repeated multiplication. An exponent is also called the power of the value.
Expression
Any combination of values and operations that can be used to show how things belong together and compare to one another.
Factor
To change two or more terms to just one term so that you can perform other processes
Operation
An action performed upon one or two numbers to produce a resulting number. Operations are addition, subtraction, multiplication, division, square root, and so on.
Solve
To find the answer. In algebra, it means to figure out what the variable stands for.
Term
A grouping together of one or more factors (variables and/or numbers), such as 4xy.
Variable
A letter that always represents an unknown number, or what you’re solving for in an algebra problem.
Coefficient
A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
Natural Numbers
‘Counting’ Numbers - 1, 2, 3, 4..
Integers
The set of whole numbers, plus negative numbers - -4, -3, -2, -1, 0, 1, 2, 3, 4…
Rational Numbers
Numbers which can be expressed as a fraction; a ratio of two integers - 1/2, 3/4, ⁻4/3, 0.5, 0.8…
Irrational Numbers
A number which cannot be expressed as a fraction; a non-terminating, non-repeating decimal - 3.14159265…
Real Numbers
Numbers that can be written as a decimal - including irrational numbers.
Surds
A square root which cannot be simplified; surds are irrational numbers.
The sum or product of two rational numbers…
…is always a rational number.
The sum or product of two irrational numbers…
…is not always an irrational number.
LCM
Lowest Common Multiple
HCF
Highest Common Factor
BIDMAS (BEDMAS)
Brackets, Indices, Division, Multiplication, Addition and Subtraction
Base
The number upon which an index works
Significant Figures
The remaining numbers - the most significant - of a simplified number
Plane Figure
Plane figures are flat two-dimensional (2D) shapes.
Soh Cah Toa
Sine Opposite over Hypotenuse / Cosine Adjacent over Hypotenuse / Tangent Opposite over Adjacent
Euler’s Number
The number e , sometimes called the natural number, or Euler’s number, is an important mathematical constant approximately equal to 2.71828. When used as the base for a logarithm, the corresponding logarithm is called the natural logarithm, and is written as ln(x).
Ln (calculator button)
Log with base ‘e’ (Euler’s Number)
Absolute Value
The actual magnitude of a numerical value or measurement, irrespective of its relation to other values.
Root vs. Log
The distinction between roots and logarithms is this: for root-taking, you specify the power and ask for the base; for logarithms, you specify the base and ask for the power (or exponent).
Ambiguous Case
The ambiguous case occurs when one uses the law of sines to determine missing measures of a triangle when given two sides and an angle opposite one of those angles (SSA)
Trigonometric Identities
The various expressions of sin, cos, tan, sec, csc, cot, etc.
Sine Period (t)
The period of the sine curve is the length of one cycle of the curve. The natural period of the sine curve is 2π.
Radian
The radian is a unit of measure for angles used mainly in trigonometry. It is used instead of degrees. Whereas a full circle is 360 degrees, a full circle is just over 6 radians. A full circle has 2π radians.
Inverse Square Law
E = I/d^2
Argand Diagram
An Argand diagram is a plot of complex numbers as points. in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis.
Element
The individual items (numbers, symbols or expressions) in a matrix are called its elements or entries. Provided that they are the same size (have the same number of rows and the same number of columns), two matrices can be added or subtracted element by element.
m x n Matrix
Rows (m) and Columns (n)
Square Matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order. Any two square matrices of the same order can be added and multiplied. Square matrices are often used to represent simple linear transformations, such as shearing or rotation.
Diagonal Matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2-by-2 diagonal matrix is, while an example of a 3-by-3 diagonal matrix is.
Identity or Unit Matrix
In linear algebra, the identity matrix of size n is the n × n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by Iₙ, or simply by I if the size is immaterial or can be trivially determined by the context.
Transpose of a Matrix
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix A by producing another matrix, often denoted by Aᵀ.
Triangular Matrix
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix. A square matrix is called lower triangular if all the entries above the main diagonal are zero. Similarly, a square matrix is called upper triangular if all the entries below the main diagonal are zero.
Determinant
In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix A is denoted det, det A, or |A|.
Coefficient Matrix
In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations
Factoring by Inspection
…
Formula for Quadratic Equations
b ± √(b^2 - (4ac)) / 2a
Simple method for Quadratic Equations
aX^2 + bX^1 + cX^0 = 0
If a = 1 and the left side of the equation = 0, find two values whose product equal either b or c and whose sum equals whichever of the two remains.
Re-write the equation as (X + x)(X + y) = 0.
The inverse of x and y are the two possible values of X.
Method for solving a 2x2 Matrix
| | c d |
a b | = A = ad - bc
Initial Point
= O = the point from which a vector calculation begins
Terminal Point
= Q = the terminal point of a vector
Resultant (vectors)
The resultant is the vector sum of two or more vectors. It is the result of adding two or more vectors together. From the Initial Point to the Terminal Point.
When displacement vectors are added, the result is a resultant displacement. But any two vectors can be added as long as they are the same vector quantity.
Paralllelogram
Parallelogram Law: This is a graphical method used for a) addition of two vectors, b) subtraction of two vectors, and c) resolution of a vector into two components in arbitrary directions.
The addition of these two vectors gives the resultant vector. The following steps are used to find the resultant vector.
Formula for calculating values in non-right-angle triangles
a / sin(A) = b / sin(B) = c sin(C)
Unit Vector
In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or “hat”, as in.
“Mathematics is like travelling the world”
When we think of travelling we think of distinct places - London, Berlin, New York, Tokyo, etc. In the same way, when we think of maths we think of distinct areas - Quadratic Equations, Geometry, Logarithms, etc.
In reality, these things are not separate; they are connected.
The Babylonians used a base-60 (sexagesimal) system, from this we still use:
60 seconds in a minute, 60 minutes in an hour. Also the 360 degrees in a circle.
Z = x + jy
Cartesian form (Mathematics) or Rectangular form (Electrical Engineering) of a complex number
Z = x + jy = r ⋅ cos θ + j ⋅ r ⋅ sin θ = r ⋅ (cos θ + j ⋅ sin θ)
r = modulus
Exponential Form of a complex number (Euler’s Formula)
cos θ + j ⋅ sin θ = e^(jθ)
De Moivre’s Formula (for exponential complex numbers)
z^(n) = [r ⋅ cos θ + j ⋅ sin θ]^(n) = r^(n) ⋅ cos (n ⋅ θ) + r^(n) ⋅ j ⋅ sin (n ⋅ θ)
Tail and Head (vectors)
The beginning point and termination point of a vector
Dot Product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used.
Cross Product
In mathematics, the cross product or vector product is a binary operation on two vectors in three-dimensional space, and is denoted by the symbol. Given two linearly independent vectors a and b, the cross product, a × b, is a vector that is perpendicular to both a and b, and thus normal to the plane containing them.
Angular Frequency
In physics, angular frequency ω (2πf) is a scalar measure of rotation rate. It refers to the angular displacement per unit time or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument of the sine function.
Minute of Arc = 1/60° = 0.016°
A minute of arc, arcminute, arc minute, or minute arc, denoted by the symbol, is a unit of angular measurement equal to of one degree. Since one degree is of a turn, one minute of arc is of a turn.
Initial Velocity
The initial velocity,vi is the velocity of the object before acceleration causes a change. After accelerating for some amount of time, the new velocity is the final velocity, vf. initial velocity = final velocity - (acceleration×time) vi = vf - at. vi = initial velocity (m/s)
Gravitational Acceleration of Earth
9.807 m/s²
Sine Rule:
sin A / a = sin B / b = sin C / c
Cosine Rule:
c = √(a^2 + b^2 - 2ab × cos C)
Linear Equations and Linear Systems
Linear Equations feature two variables, e.g. 2X +5Y = 10. Linear Systems feature three variables, e.g. 2X + 5Y + 8Z = 10.
Linear equations can be solved graphically by sketching both equations provided on the x, y axis and finding the intersecting point. Or use substitution.
Linear systems can be solved by subtracting the equations from each other, or they can be solved using matrices.
Scalars and Vectors
Scalars or ‘Scalar Quantities’ are defined by only a single value, such as temperature, mass, or distance.
Vectors are values which have both magnitude and direction. Such as a force applied downwards, or the speed of a car in a given direction.
Nose and Tail
Respectively, the arrow and tail ends of a drawn vector
Quadratic and Cubic Equations
Quadratic equations (ax^2 + bx + c = 0) and Cubic Equations (ax^3 + bx^2 + xc + d = 0) can be solved using Matlab or using their respective formulas. Quadratic equations can also be solved using factorisation.
