IB Mathematics: Number and Algebra Review

These vocabulary flashcards provide key definitions, notations, and formulas for IB Mathematics Analysis and Approaches, covering number sets, sequences, financial math, logic, proofs, and complex numbers.

Natural Numbers ($N$)

The set of non-negative integers represented as $\{0, 1, 2, 3, 4, \dots\}$.

Integers ($Z$)

The set containing all whole numbers and their negatives: $\{0, \pm1, \pm2, \pm3, \dots\}$.

Rational Numbers ($Q$)

Numbers that can be expressed as a fraction $\frac{a}{b}$ where $a, b \in Z$ and $b \neq 0$.

Real Numbers ($R$)

The complete set of numbers comprising both rational and irrational values.

Significant Figures (s.f.)

A rounding method where counting begins from the first non-zero digit of a number.

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

A way of writing numbers where $1 \leq a < 10$ and $k$ is an integer.

Sequence

An ordered list of numbers, often denoted as $u_1, u_2, u_3, \dots$.

Series

The total sum of the terms in a sequence, which can be finite ($S_n$) or infinite ($S_{\infty}$).

Sigma Notation ($\sum_{n=1}^{k} u_n$)

A symbolic representation used to indicate the sum of terms $u_n$ from an initial index to a final index.

Recursive Relation

A method of defining a sequence by providing the first term(s) and a formula for $u_{n+1}$ based on preceding terms.

Arithmetic Sequence (A.S.)

A sequence in which the difference between any two consecutive terms is a constant value, $d$.

Arithmetic General Formula

The formula used to find the nth term of an arithmetic sequence: $u_n = u_1 + (n - 1)d$.

Arithmetic Sum Formula

Calculated as $S_n = \frac{n}{2}(u_1 + u_n)$ or $S_n = \frac{n}{2}[2u_1 + (n - 1)d]$.

Geometric Sequence (G.S.)

A sequence where the ratio between any two consecutive terms remains constant, denoted as $r$.

Geometric General Formula

The formula used to find the nth term of a geometric sequence: $u_n = u_1 r^{n-1}$.

Geometric Sum Formula

The sum of the first $n$ terms is $S_n = \frac{u_1(r^n - 1)}{r - 1}$ or $S_n = \frac{u_1(1 - r^n)}{1 - r}$, where $r \neq 1$.

Infinite Geometric Series

A series that converges when $|r| < 1$, with a total sum given by $S_{\infty} = \frac{u_1}{1 - r}$.

Future Value ($FV$)

The amount an investment grows to over $n$ periods at rate $r\%$, calculated as $FV = PV(1 + \frac{r}{100})^n$.

Compound Interest (k periods)

Interest calculated periodically within a year, using the formula $FV = PV(1 + \frac{r}{100k})^{kn}$.

Real Value (RV)

The value of an investment adjusted for inflation ($a\%$), calculated as $RV = \frac{FV}{(1 + \frac{a}{100})^n}$.

Factorial ($n!$)

The product of all positive integers up to $n$, where $1! = 1$ and $0! = 1$.

Binomial Coefficient ($nCr$)

The number of ways to choose $r$ items from $n$, calculated as $\binom{n}{r} = \frac{n!}{r!(n - r)!}$.

Binomial Theorem

The formal expansion of $(a + b)^n$ into a sum of terms involving powers of $a$ and $b$ and coefficients from Pascal's triangle.

Identity ($\equiv$)

A mathematical statement that remains true for all possible values of the variables involved.

Converse

The statement formed by reversing the hypothesis and conclusion of an implication: if $B$ then $A$.

Contrapositive

A statement equivalent to the original implication: if not $B$ then not $A$.

Proof by Contradiction

A method where one assumes the negation of the statement to be proven and derives a logical inconsistency.

The Pigeonhole Principle

If $n+1$ pigeons are placed in $n$ holes, at least one hole must contain at least 2 pigeons.

Mathematical Induction

A three-step proof technique: testing a base case ($n=1$), assuming it true for $n=k$, and proving it for $n=k+1$.

Consistent System

A system of linear equations that possesses at least one solution.

Gaussian Elimination

A methodology using row operations on an augmented matrix to reduce it to a form where variables can be solved through back-substitution.

Complex Number ($z$)

A number of the form $x + yi$, where $x$ is the real part and $y$ is the imaginary part.

Imaginary Unit ($i$)

A number defined such that $i^2 = -1$.

Complex Conjugate ($\bar{z}$)

For a number $z = x + yi$, the conjugate is defined as $\bar{z} = x - yi$.

Modulus ($|z|$)

The distance of a complex number from the origin in the complex plane: $|z| = \sqrt{x^2 + y^2}$.

Fundamental Theorem of Algebra

States that every polynomial of degree $n \geq 1$ has exactly $n$ complex roots.

Conjugate Root Theorem

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

Polar Form ($r \text{cis}(\theta)$)

Representing a complex number using its modulus $r$ and argument $\theta$ as $r(\cos(\theta) + i\sin(\theta))$.

Euler's Form

Expressing a complex number as $z = re^{i\theta}$, identifying $e^{i\theta}$ with $\cos(\theta) + i\sin(\theta)$.

De Moivre’s Theorem

States that for any integer $n$, $[r(\cos(\theta) + i\sin(\theta))]^n = r^n(\cos(n\theta) + i\sin(n\theta))$.

n-th Roots of Unity

The $n$ distinct solutions to the equation $z^n = 1$, which divide the unit circle into $n$ equal arcs.

Sum of n-th Roots of Unity

The property that the sum of all distinct solutions to $z^n = 1$ is always equal to 0.