Number & Algebra: notes and practice questions

Scientific notation expresses numbers as $a \times 10^k$. Conditions for scientific notation: $1 \le a < 10$ and $k \in \mathbb{Z}$.

A sequence is an ordered set of numbers; terms are denoted by $u_n$. A series is the sum of terms in a sequence; the sum of the first $n$ terms is $S_n$.

A geometric sequence has a common ratio, $r$, between consecutive terms ($r = u_n / u_{n-1}$). Sequence behavior: increasing if $r > 1$, decreasing if $0 < r < 1$, alternating if $r < 0$.

Simple Interest: Interest only on initial investment (principal). Compound Interest: Interest on initial investment and accumulated interest; exponential growth.

An exponent is a power a base is raised to. A logarithm is its inverse, calculating the power. Fundamental relationship: $a^x = b \iff \log_a(b) = x$ (where $a > 0, b > 0, a \neq 1$).

IB exam default accuracy: 3 significant figures (3 s.f.) for non-exact answers. Use exact values (e.g., $\pi$, $\sqrt{2}$) in working to avoid rounding errors.

Amortisation: Process of repaying a loan over a fixed period. Annuity: Fixed sum of money paid at specified intervals from an initial investment.

A linear equation is of degree 1, generally written as $ax + by + c = 0$. A system of linear equations is a set of two or more linear equations sharing variables.

A logarithm is the inverse of an exponent: $a^x = b \iff \log_a(b) = x$ (where $a > 0, b > 0, a \neq 1$). Common logarithm (base 10): $\log x = \log_{10}(x)$.

Exponent (Index): A power a base is raised to. Rational Exponent: A fractional power where the denominator is a root and the numerator is a standard power.

A geometric sequence is divergent if terms increase/decrease away from zero; sum approaches $\pm \infty$. A geometric sequence is convergent if terms get closer to zero; sum approaches a finite limit.

The imaginary unit: $i = \sqrt{-1}$, so $i^2 = -1$. Square roots of negative numbers: $\sqrt{-k} = i\sqrt{k}$ for $k > 0$.

Modulus of $z = x + iy$ is $r = |z| = \sqrt{x^2 + y^2}$, representing distance from origin. Argument of $z$ is $\theta = \text{arg } z$, the angle with the positive real axis (counter-clockwise, radians, usually $-\pi < \theta \le \pi$ or $0 \le \theta < 2\pi$).

Matrix order: $m \times n$ (rows x columns); elements: $a_{i,j}$. Square matrix: $m=n$.

An eigenvector $x$ of matrix $A$ changes only in scale by factor $\lambda$ (eigenvalue): $Ax = \lambda x$. Characteristic polynomial: $p(\lambda) = \det(\lambda I - A)$.