Introduction to Algebra and Calculus (MATH 173)

Comprehensive vocabulary flashcards covering number operations, linear and quadratic equations, polynomials, matrix algebra, determinants, differentiation, and integration from MATH 173.

Natural Numbers

All positive numbers starting from $1$, denoted by the letter "$n$" in arithmetic expressions (zero is not included).

Whole Numbers

Numbers that include all natural numbers as well as zero ($0, 1, 2, 3\dots$), containing no decimal or fractional parts and never taking negative values.

Integers

Numbers representing whole units of positive and negative values as well as zero ($\dots -2, -1, 0, 1, 2\dots$), with no decimal or fractional parts.

Fractions

Parts of whole numbers that indicate one integer is being divided by another integer.

Decimals

Numbers based on the decimal or $10$s system that use a dot or point to separate the ones place from the tenths place.

BEDMAS

An order of operations rule indicating the sequence: Brackets, Exponents, Division and Multiplication (left to right), and Addition and Subtraction (left to right).

BODMAS

A variation of the order of operations rule where "O" stands for Orders (including powers/exponents and roots) instead of Exponents.

Single Variable Linear Equation

An equation with only one unknown quantity whose variable has a maximum power of one and forms a straight line when plotted on a graph.

Simultaneous Linear Equations

A set of two or more equations with common variables that are solved together using methods such as the elimination method or substitution method.

Quadratic Formula

The formula used to solve quadratic equations of the form $ax^2 + bx + c = 0$ ($a \neq 0$), defined as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Gradient (Slope)

Represented by $m$ in the straight line equation $y = mx + c$, it is the ratio of change in $y$ to change in $x$, calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

$y$-Intercept

Represented by $c$ in the linear equation $y = mx + c$, it is the coordinate point where a straight line crosses the $y$-axis.

Monomial

An algebraic expression that is either a real number, a variable, or a product of real numbers and variables.

Polynomial

An algebraic expression that is a sum of terms, where each term contains only variables with whole number exponents and real number coefficients.

Standard Form of a Polynomial

The arrangement of a polynomial such that its terms are written in descending order of exponents from left to right.

Degree of a Term

The exponent of the variable in a single term, or the sum of the exponents of all variables within that term when multiple variables are present.

Degree of a Polynomial

The largest degree among any of the individual terms in the polynomial.

Binomial

A polynomial classification designating an expression that contains exactly two terms.

Trinomial

A polynomial classification designating an expression that contains exactly three terms.

Like Terms

Terms that possess the identical variable(s) raised to the identical exponents, though they may carry different numerical coefficients.

Roots of a Polynomial

The numerical solutions obtained when a polynomial expression or equation is set equal to zero.

Zeros of a Polynomial Function

The values of $x$ that satisfy $f(x) = 0$, which correspond to the $x$-intercepts on the graph of the polynomial function.

Factor Theorem

A theorem stating that for a polynomial function $f(x)$, if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$.

Remainder Theorem

A theorem stating that when a polynomial $f(x)$ is divided by a linear factor $(x - a)$, the remainder is equal to $f(a)$.

Matrix

A rectangular array of variables or constants organized in horizontal rows and vertical columns enclosed within brackets.

Element (Matrix)

Each individual value (either a number or a constant) located inside a matrix.

Dimension (Matrix)

The size or order of a matrix, expressed as the number of rows by the number of columns ($m \times n$).

Row Matrix

A matrix consisting of only one horizontal row.

Column Matrix

A matrix consisting of only one vertical column.

Square Matrix

A matrix that has an equal number of rows and columns.

Null Matrix

A matrix in which every single element is zero.

Diagonal Matrix

A square matrix in which all non-diagonal elements are zero, leaving only the main diagonal with non-zero values.

Identity Matrix (Unit Matrix)

A diagonal matrix in which every diagonal element is equal to one, denoted by $I$.

Equal Matrices

Two matrices that share identical dimensions with every corresponding element being equal in value.

Transpose of a Matrix

A matrix $A^T$ formed by interchanging the rows of matrix $A$ into columns such that each entry satisfies $b_{ij} = a_{ji}$.

Symmetric Matrix

A square matrix that is equal to its own transpose, satisfying $A^T = A$.

Determinant

A single scalar value calculated by multiplying and combining the elements of a square matrix in a specific systematic way.

Cramer's Rule

A method for solving systems of linear simultaneous equations by utilizing determinants of the coefficient and modified matrices.

Minor of a Matrix

The determinant of the submatrix resulting from crossing out or deleting the row and column of a given element in a square matrix.

Cofactor of a Matrix

The minor of a matrix element multiplied by a sign factor, mathematically defined by $C_{ij} = (-1)^{i+j} M_{ij}$.

Adjoint of a Matrix

The transpose of the cofactor matrix of a given square matrix, denoted by $\text{Adj}(P) = [C_{ij}]^T$.

Non-Singular (Invertible) Matrix

A square matrix that possesses an inverse, defined by having a determinant not equal to zero ($\det(A) \neq 0$).

Singular (Non-Invertible) Matrix

A square matrix that does not have an inverse, defined by having a determinant equal to zero ($\det(A) = 0$).

Inverse of a Matrix

A matrix $A^{-1}$ satisfying $A A^{-1} = A^{-1} A = I$, calculated as $A^{-1} = \frac{1}{\det(A)} \text{Adj}(A)$.

Differentiation

The mathematical process of finding the derivative or rate of change of a function.

Derivative

The result of differentiation that measures how fast a function is changing, representing the slope of the curve or tangent line.

Limit

The value $L$ that a function $f(x)$ approaches as the independent variable $x$ gets closer and closer to a given value $a$.

Theorem

A mathematical statement that has been proven to be true through logical demonstration rather than opinion or guessing.

First Principles of Differentiation

The foundational limit definition of a derivative, expressed as $f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$.

Product Rule

The differentiation rule stating that if $y = uv$, where $u$ and $v$ are both functions of $x$, then $\frac{dy}{dx} = u\frac{dv}{dx} + v\frac{du}{dx}$.

Quotient Rule

The differentiation rule stating that if $y = \frac{u}{v}$, then $\frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$.

Integration

The mathematical operation that acts as the reverse of differentiation to find the original function from a given derivative or gradient function.

Definite Integral

An integral evaluated between two specified limits ($\int_a^b y\,dx$), representing the value of the anti-derivative at $x = b$ minus its value at $x = a$.