Comprehensive Mathematics and Engineering Review Flashcards

Flashcards covering key mathematical concepts, formulas, and terminology from the lecture notes.

Cardinal Numbers

Numbers used for counting, such as $1, 2, 3, \dots$

Ordinal Numbers

Numbers indicating position or order, such as First, Second, Third, etc.

Roman Numeral I

Represents the numeric value $1$

Roman Numeral V

Represents the numeric value $5$

Roman Numeral X

Represents the numeric value $10$

Roman Numeral L

Represents the numeric value $50$

Roman Numeral C

Represents the numeric value $100$

Roman Numeral D

Represents the numeric value $500$

Roman Numeral M

Represents the numeric value $1000$

Roman Numeral Multiplier $|X|$

Indicates multiplication of the Roman numeral value by $1000$

Roman Numeral Multiplier $\overline{X}$

Indicates multiplication of the Roman numeral value by $10000$

Roman Numeral Multiplier $[X]$

Indicates multiplication of the Roman numeral value by $1000000$

Natural Numbers

Positive counting numbers starting from $1$ ($1, 2, 3, \dots$)

Integers

Whole numbers including negative numbers, zero, and positive numbers ($\dots, -1, 0, 1, \dots$)

Rational Numbers

Numbers that can be expressed as a fraction or terminating/repeating decimal ($0.5$, $\frac{2}{3}$, $0.333\dots$)

Irrational Numbers

Real numbers that cannot be expressed as a simple fraction (e.g., $\sqrt{2}$, $\pi$, $e$)

Imaginary Numbers

Numbers expressed as the product of a real unit and $i$, $j$, or $\sqrt{-1}$

Complex Numbers

Numbers composed of a real part and an imaginary part, written as $a + bi$

Factorial

The product of all positive integers less than or equal to $n$, defined as $n! = n(n-1)\dots 3\cdot 2\cdot 1$

Addend

Any of the numbers that are added together in an addition equation ($A$ and $B$ in $A + B = S$)

Sum

The total result of adding two or more numbers ($S$ in $A + B = S$)

Minuend

The quantity from which another quantity is subtracted ($A$ in $A - B = D$)

Subtrahend

The quantity that is to be subtracted from another ($B$ in $A - B = D$)

Difference

The result of a subtraction operation ($D$ in $A - B = D$)

Multiplicand

The number that is to be multiplied by another ($A$ in $A \times B = P$)

Multiplier

The factor by which the multiplicand is multiplied ($B$ in $A \times B = P$)

Product

The result of a multiplication operation ($P$ in $A \times B = P$)

Dividend

The number that is being divided ($A$ in $A / B = Q$)

Divisor

The number by which the dividend is divided ($B$ in $A / B = Q$)

Quotient

The result of a division operation ($Q$ in $A / B = Q$)

Remainder Theorem Step 1

Equate the linear divisor (e.g., $x + 1$) to zero and solve for $x$ (e.g., $x = -1$)

Remainder Theorem Step 2

Substitute the calculated value of $x$ into the target polynomial equation to evaluate the remainder

Standard Form of Quadratic Equation

$$Ax^2 + Bx + C = 0$$

Quadratic Formula

$$x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$$

Sum of Roots of Quadratic Equation

$$r_1 + r_2 = -\frac{B}{A}$$

Product of Roots of Quadratic Equation

$$r_1 r_2 = \frac{C}{A}$$

Condition for Equal and Opposite Roots

When $B = 0$, the roots satisfy $r_1 = -r_2$

Binomial Theorem r-th Term Formula

$$r^{\text{th}}\text{ term} = {}_n C_{r-1} x^{n-r+1} y^{r-1}$$

Binomial Theorem Term with $y^r$

$$y^r = {}_n C_r x^{n-r} y^r$$

Sum of Coefficients in Binomial Expansion

$$\text{Sum of Coef.} = (\text{Coef. of } x + \text{Coef. of } y)^n$$

Sum of Exponents in Binomial Expansion

$$\text{Sum of Exp.} = n(n - 1)$$

Equivalences of 1 Revolution

$$1\text{ Rev} = 360^\circ = 2\pi = 6400\text{ mils} = 6400\text{ gons}$$

Logarithm Product Rule

$$\log(xy) = \log(x) + \log(y)$$

Logarithm Quotient Rule

$$\log\left(\frac{x}{y}\right) = \log(x) - \log(y)$$

Logarithm Power Rule

$$\log(x^n) = n\log(x)$$

Logarithm Base Change Rule 1

$$\log_b(x) = \frac{\log(x)}{\log(b)}$$

Logarithm Base Change Rule 2

$$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$$

Age Problems Time Tenses Notation

5 years ago: $x - 5$; Present: $x$; 3 years hence: $x + 3$

Work Rate Formula

$$r = \frac{1}{t}$$

Combined Work Formula

$$\frac{1}{a} + \frac{1}{b} = \frac{1}{T}$$

Mixture Problem Diagram Representation

Represented by rectangles/squares for components: $A\text{ L} + B\text{ L} = C\text{ L}$ with concentrations $X\%$, $Y\%$, $Z\%$

Mixture Concentration Equation

$A\left(\frac{X}{100}\right) + B\left(\frac{Y}{100}\right) = C\left(\frac{Z}{100}\right)$ where $A + B = C$

3-Digit Number Representation

$$\text{Number} = h(100) + t(10) + u$$

2-Digit Number Representation

$$\text{Number} = t(10) + u$$

Clock Problem Hand Relation (Minute Hand)

$$HH = \frac{MH}{12}$$

Clock Problem Hand Relation (Second Hand)

$$HH = \frac{SH}{720}$$

Arithmetic Progression n-th Term Formula

$$a_n = a_1 + (n - 1)d$$

Arithmetic Progression Sum Formula 1

$$S = \frac{n}{2}(a_1 + a_n)$$

Arithmetic Progression Sum Formula 2

$$S = \frac{n}{2}[2a_1 + (n - 1)d]$$

Geometric Progression n-th Term Formula

$$a_n = a_1 r^{n-1}$$

Geometric Progression Sum Formula

$$S = \frac{a_1(r^n - 1)}{r - 1}$$

Infinite Geometric Progression ($r > 1$)

$$S = \infty$$

Infinite Geometric Progression ($r < 1$)

$$S = \frac{a_1}{1 - r}$$

Direct Variation Equation

$$x = ky$$

Inverse Variation Equation

$$x = k\left(\frac{1}{y}\right)$$

Proportionality Constant

The constant $k$ in variation equations $x = ky$ or $x = \frac{k}{y}$

Venn Diagram Subset-Only Calculation Step 1

Find the amount of people that belongs only to each specific subset (e.g., Math only: $50 - 30 - 28 + 12 = 4$)

Venn Diagram Remaining Calculation Step 2

Find the remaining amount of people (e.g., $28 + 30 + 26 - 2(12) = 60$)

Venn Diagram Total Students Step 3

Add all individual subset and intersection counts to get total students

Permutation Formula

$${}_n P_r = \frac{n!}{(n - r)!}$$

Combination Formula

$${}_n C_r = \frac{n!}{(n - r)! r!}$$

Probability of Event Formula

$P_E = \frac{S}{T}$ where $S$ is successful outcomes and $T$ is total outcomes

Probability of Complementary Event

$$P_{\text{not } E} = 1 - P_E$$

Probability Addition Rule (E or F)

$$P_{E \text{ or } F} = P_E + P_F$$

Probability Multiplication Rule (E & F)

$$P_{E \text{ \& } F} = P_E P_F$$

Binomial Probability Distribution Formula

$P = {}_n C_r p^r q^{n-r}$ where $p$ is probability of success, $q = 1-p$ is probability of failure, $n$ is trials, $r$ is successes

Sine Wave Equation

$$y(x,t) = A\sin(kx - \omega t + \phi)$$

Sine Wave Amplitude Parameter

Represented by $A$ in $y(x,t) = A\sin(kx - \omega t + \phi)$

Sine Wave Angular Frequency Parameter

Represented by $\omega$

Sine Wave Period Formula

$$T = \frac{2\pi}{\omega}$$

Sine Wave Frequency Formula

$$f = \frac{1}{T}$$

Sine Wave Wavelength Formula

$$\lambda = \frac{2\pi}{k}$$

Sine Wave Initial Phase Shift Parameter

Represented by $\phi$

Rose Curve Equation

$$r = 2\cos(k\theta)$$

Rose Curve Petals (k is odd)

The rose curve has $k$ petals

Rose Curve Petals (k is even)

The rose curve has $2k$ petals

Rose Curve Petals (k is half-integer)

The rose curve has $4k$ petals (e.g., $k = \frac{5}{2}$ results in 10 petals)

Set Definition and Symbol

Collection of elements, denoted by symbol $\{\}$

Intersection Definition and Symbol

Elements belonging to both set $A$ and set $B$, symbol $A \cap B$

Union Definition and Symbol

All objects in set $A$ and set $B$, symbol $A \cup B$

Subset Definition and Symbol

$A$ is contained within $B$, symbol $A \subseteq B$

Superset Definition and Symbol

$A$ contains set $B$, symbol $A \supseteq B$

Equal Sets Definition and Symbol

$A$ is equal to $B$, symbol $A = B$

Complement Definition and Symbol

All elements that do not belong to set $A$, symbol $A'$

Inequality Interval Parenthesis Rule

$\text{()}$ indicates that the boundary number is NOT a solution to the equation

Inequality Interval Bracket Rule

$\text{[]}$ indicates that the boundary number IS a solution to the equation

Odd Function Property

$f(-x) \neq f(x)$ (e.g., $f(x) = x - 3$ yields $-5 \neq -1$ when evaluating $x = -2$ and $2$)

Even Function Property

$f(-x) = f(x)$ (e.g., $f(x) = x^2$ yields $4 = 4$ when evaluating $x = -2$ and $2$)

Inverse Function Step 1

Change $f(x)$ to $y$ (e.g., $y = \frac{x + 3}{2}$)

Inverse Function Step 2

Equate and solve the equation in terms of $x$ (e.g., $x = 2y - 3$)