IB^® Math Analysis & Approaches HL/SL

THE BIG PICTURE. Topic 1 builds the algebraic foundation for everything that follows. SL focuses on sequences and series, exponents and logarithms, and the binomial theorem. SL also covers scientific notation and an introduction to proof. HL extends to counting principles, complex numbers, formal proof (induction/contradiction/counter-example), systems of linear equations, and partial fractions. Strong fluency here pays dividends across all five topics: every other topic uses these tools.

Arithmetic: each term differs by a constant common difference $d$:

• $u_n = u_1 + (n - 1) d$
• $S_n = \dfrac{n}{2}(2 u_1 + (n - 1) d) = \dfrac{n}{2}(u_1 + u_n)$

Geometric: each term multiplied by a constant common ratio $r$:

• $u_n = u_1 \cdot r^{n-1}$
• $S_n = \dfrac{u_1 (1 - r^n)}{1 - r}$ for $r \neq 1$
• Sum to infinity (only when $|r| < 1$): $S_\infty = \dfrac{u_1}{1 - r}$

Sigma notation compresses long sums: $\sum_{i=1}^n u_i = u_1 + u_2 + \cdots + u_n$.

Compound interest is a geometric application: $\text{FV} = \text{PV} \left(1 + \dfrac{r}{100k}\right)^{kn}$, with $k$ compounding periods per year for $n$ years at nominal rate $r\%$. Annual depreciation uses the same form with negative $r$.

Exponent laws: $a^m \cdot a^n = a^{m+n}$, $\dfrac{a^m}{a^n} = a^{m-n}$, $(a^m)^n = a^{mn}$, $a^0 = 1$, $a^{-n} = \dfrac{1}{a^n}$, $a^{1/n} = \sqrt[n]{a}$.

Definition of logarithm: $a^x = b \;\Leftrightarrow\; x = \log_a b$ (with $a > 0$, $b > 0$, $a \neq 1$).

Logarithm laws mirror exponent laws

• $\log_a (xy) = \log_a x + \log_a y$
• $\log_a \dfrac{x}{y} = \log_a x - \log_a y$
• $\log_a x^m = m \log_a x$
• $\log_a a = 1$, $\log_a 1 = 0$
• Change of base: $\log_a x = \dfrac{\log_b x}{\log_b a}$
• Natural log: $\ln x = \log_e x$, with $e \approx 2.71828$

Solving exponential equations: take logs of both sides; isolate the variable. Always check that the argument is positive.

• Pascal's triangle generates binomial coefficients for small $n$.
• The $(r+1)$th term is $\binom{n}{r} a^{n-r} b^r$: useful for "find the term containing $x^k$" questions.
• HL extension: the theorem extends to negative and rational $n$ for $|b/a| < 1$: $(1 + x)^n = 1 + nx + \dfrac{n(n-1)}{2!} x^2 + \cdots$

Cartesian form: $z = a + bi$ with $i = \sqrt{-1}$, so $i^2 = -1$. $\text{Re}(z) = a$, $\text{Im}(z) = b$.

Modulus: $|z| = \sqrt{a^2 + b^2}$. Argument: $\arg(z) = \theta$ where $\tan \theta = b/a$ (always check the quadrant).

Polar (modulus-argument) form: $z = r(\cos\theta + i\sin\theta) = r\,\text{cis}\,\theta$.

Exponential (Euler) form: $z = r e^{i\theta}$, leveraging Euler's identity $e^{i\theta} = \cos\theta + i\sin\theta$.

De Moivre's Theorem: $z^n = r^n (\cos n\theta + i \sin n\theta) = r^n e^{i n\theta}$. Used for powers and $n$th roots of complex numbers; the $n$ roots of $z$ are equally spaced by $\dfrac{2\pi}{n}$ on a circle of radius $|z|^{1/n}$.

Example. Find the four 4th roots of $16 i$. Write $16 i = 16\, e^{i(\pi/2 + 2\pi k)}$ for integer $k$. 4th roots: $z_k = 16^{1/4}\, e^{i(\pi/8 + \pi k/2)} = 2\, e^{i(\pi/8 + \pi k/2)}$ for $k = 0, 1, 2, 3$. Four points equally spaced on a circle of radius 2, starting at angle $\pi/8$.

Conjugate $z^* = a - bi$. Properties: $z + z^* = 2a$ (real), $z \cdot z^* = |z|^2$ (real, non-negative). Used to rationalise complex denominators.

Fundamental theorem of algebra: every polynomial of degree $n$ with real or complex coefficients has exactly $n$ roots (counting multiplicity) in $\mathbb{C}$. Real polynomial roots come in conjugate pairs: if $p + qi$ is a root, so is $p - qi$.

SL students are introduced to the formal concept of proof: what a proof is, and simple deductive proof (numerical and algebraic; laying out a clear LHS → RHS argument with correct equality/identity notation). HL extends this to the three formal methods below: induction, contradiction, and counter-example.

Methods (direct/deductive at SL; induction, contradiction, counter-example are HL):

• Direct (deductive) proof: assume the hypothesis, deduce the conclusion through valid algebraic steps.
• Proof by contradiction: assume the negation of what you want to prove, derive a contradiction. Classic example: $\sqrt{2}$ is irrational.
• Proof by induction: for statements indexed by $n \in \mathbb{Z}^+$:

1. Base case: verify $P(1)$ is true. 2. Inductive step: assume $P(k)$ is true; prove $P(k+1)$ follows. 3. Conclusion: by induction, $P(n)$ is true for all $n \in \mathbb{Z}^+$.

• Disproof by counter-example: one specific case where the statement fails is enough to disprove a universal claim.

Induction is heavily tested on summation formulas, divisibility, inequalities, and De Moivre's theorem (whose induction proof is explicitly on the HL syllabus). Matrices are not part of Math AA.

Useful for integration and Maclaurin series.

EXAM CONNECTIONS.

• Paper 1 (no GDC): algebraic manipulation of logs, geometric sums, induction proofs (HL), De Moivre (HL).
• Paper 2 (with GDC): finance/compound-interest problems, larger numerical sums, complex roots via GDC.
• Paper 3 (HL only): extended investigations frequently use induction, counting, or complex-number identities.