Topic 1: Number and Algebra

Vocabulary practice flashcards covering fundamental definitions, theorems, formulas, and concepts from IB Mathematics Topic 1: Number and Algebra.

Natural Numbers ($\mathbb{N}$)

The set of non-negative whole counting numbers, defined as $\{0, 1, 2, 3, 4, \dots\}$.

Rational Numbers ($\mathbb{Q}$)

The set of numbers that can be written as fractions of integers, defined as $\{\frac{a}{b} : a, b \in \mathbb{Z}, b \neq 0\}$.

Scientific Form ($a \times 10^k$)

A standard way of writing numbers where $1 \le a < 10$ and $k \in \mathbb{Z}$, obtained by placing the decimal point immediately after the first non-zero digit.

Sequence

An ordered list of numbers with terms arranged in a definite order, denoted as $u_1, u_2, u_3, \dots, u_n$.

Partial Sum ($S_n$)

The sum of the first $n$ terms of a sequence, given by $S_n = u_1 + u_2 + u_3 + \dots + u_n$.

Arithmetic Sequence

A sequence in which the difference between any two consecutive terms is constant, with general term $u_n = u_1 + (n - 1)d$.

Common Difference ($d$)

The fixed constant added to each term to obtain the next term in an arithmetic sequence, calculated as $d = u_n - u_{n-1}$.

Geometric Sequence

A sequence in which the ratio between any two consecutive terms is constant, with general term $u_n = u_1 r^{n-1}$.

Common Ratio ($r$)

The fixed multiplier between consecutive terms in a geometric sequence, calculated as $r = \frac{u_n}{u_{n-1}}$.

Sum to Infinity ($S_\infty$)

The sum of all terms of an infinite geometric sequence, which converges to $S_\infty = \frac{u_1}{1 - r}$ if and only if $-1 < r < 1$.

Amortization

The process of an investment's future value decreasing to zero over time when regular withdrawals ($PMT$) exceed the interest earned.

Factorial ($n!$)

The product of all positive integers from $1$ up to $n$, defined as $n! = 1 \times 2 \times 3 \times \dots \times n$, where $0! = 1$.

Binomial Coefficient ($^nC_r$ or $\binom{n}{r}$)

The number of ways to choose $r$ items from a set of $n$ items without regard to order, given by $\frac{n!}{r!(n - r)!}$.

Binomial Theorem

The algebraic formula for expanding powers of a binomial: $(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$, where the general term is $\binom{n}{r} a^{n-r} b^r$.

Mathematical Identity ($\equiv$)

A mathematical statement of equality that remains true for all possible values of the variables involved, unlike an equation which holds only for specific values.

Contrapositive

The logically equivalent statement formed by negating and swapping the hypothesis and conclusion of an implication: the contrapositive of "If $A$, then $B$" ($A \Rightarrow B$) is "If not $B$, then not $A$" ($\text{not } B \Rightarrow \text{not } A$).

Proof by Contradiction

A method of proof where one assumes the negation of the proposition to be proven and demonstrates that this assumption leads logically to an impossible statement.

Pigeonhole Principle

The principle stating that if $n + 1$ pigeons are placed into $n$ pigeonholes, there exists at least one pigeonhole containing at least $2$ pigeons.

Mathematical Induction

A proof method for statements $P(n)$ for $n \ge 1$ consisting of three steps: proving $P(1)$ is true, assuming $P(k)$ is true, and proving $P(k+1)$ is true based on that assumption.

Consistent System

A system of simultaneous linear equations that possesses at least one solution (either a unique solution or infinitely many solutions).

Inconsistent System

A system of simultaneous linear equations that has no solution.

Gaussian Elimination

A systematic method for solving systems of linear equations by performing row operations on an augmented matrix to eliminate coefficients below the main diagonal, followed by back substitution.

Imaginary Unit ($i$)

A mathematical unit defined by the property $i^2 = -1$ (or $i = \sqrt{-1}$).

Complex Number

A number of the form $z = x + yi$, where $x, y \in \mathbb{R}$, with real part $\text{Re}(z) = x$ and imaginary part $\text{Im}(z) = y$.

Complex Conjugate ($\bar{z}$)

For a complex number $z = x + yi$, the number defined as $\bar{z} = x - yi$, representing a reflection across the real axis.

Modulus of a Complex Number ($|z|$)

The distance of a complex number $z = x + yi$ from the origin on the complex plane, defined as $|z| = \sqrt{x^2 + y^2}$.

Fundamental Theorem of Algebra

The theorem stating that every polynomial of degree $n \ge 1$ has exactly $n$ roots in the complex field $\mathbb{C}$.

Conjugate Roots Theorem

The theorem stating that if a polynomial with real coefficients has a non-real complex root $z = a + bi$, then its conjugate $\bar{z} = a - bi$ is also a root.

Argument ($\arg(z)$)

The angle $\theta$ formed between the positive real axis and the vector representing $z$ on the complex plane, where the principal argument satisfies $-\pi < \theta \le \pi$.

Polar Form

The trigonometric representation of a complex number given by $z = r(\cos(\theta) + i\sin(\theta))$ (or $z = r\operatorname{cis}(\theta)$), where $r = |z|$ and $\theta = \arg(z)$.

Euler's Form

The exponential representation of a complex number given by $z = re^{i\theta}$, based on Euler's identity $e^{i\theta} = \cos(\theta) + i\sin(\theta)$.

De Moivre's Theorem

The theorem stating that for any complex number $z = r\operatorname{cis}(\theta)$ and any integer $n$, $z^n = r^n\operatorname{cis}(n\theta) = r^n(\cos(n\theta) + i\sin(n\theta))$.

$n$-th Roots of Unity

The $n$ distinct complex solutions to $z^n = 1$, given by $z_k = \operatorname{cis}\left(\frac{2k\pi}{n}\right)$ for $k = 0, 1, \dots, n - 1$, which divide the unit circle into $n$ equal parts and always sum to $0$.