Number & Algebra: notes and practice questions

A method to express very large or small numbers in the form: $a \times 10^n$.

### Arithmetic Sequences. An arithmetic sequence has a common difference $d$ between consecutive terms.

Geometric Sequence: A sequence with a common ratio, $r$, between consecutive terms. The sequence can be increasing ($r>1$), decreasing ($0 < r < 1$), or alternating ($r < 0$).

Compound Interest: Interest is paid on both the initial investment and any interest already earned. It can be calculated using a GDC's finance solver or the formula: $FV = PV \times \left(1 + \frac{r}{100k}\right)^{kn}$.

Standard Form (Scientific Notation): Expresses numbers in the form $a \times 10^k$, where $1 \le a < 10$ and $k$ is an integer ($k \in \mathbb{Z}$). Laws of Indices (Exponents): Rules used to simplify and manipulate expressions involving exponents. They require terms to have the same base and are not in the formula booklet. Key laws include:

A proof is a series of logical steps used to show that a result is true for all specified values, usually algebraically. To prove a statement, you must show that the left-hand side (LHS) is the same as the right-hand side (RHS).

A logarithm is the inverse of exponentiation. If $a^x = b$, then $\log_a b = x$. Key rules:

The binomial theorem provides a way to expand $(a+b)^n$ for positive integer $n$: $$.

Product Principle: Use for sequential events (AND). Sum Principle: Use for mutually exclusive choices (OR).

Extending the binomial theorem to expansions of the form $(1+x)^n$ where $n$ is a rational number. The binomial expansion is given by:

Begin by comparing the degrees of the numerator and denominator. If the numerator's degree is greater than or equal to the denominator's, perform polynomial long division first. Factorise the denominator completely.

Complex numbers are expressed in Cartesian form $z = x + iy$, with operations like addition, subtraction, multiplication, and division defined. The complex conjugate $z^* = x - iy$ is crucial. Modulus $|z|$ is the distance from the origin, and argument $\theta$ is the angle with the positive real axis.

Conjugate Root Theorem: Complex roots of polynomials with real coefficients occur in conjugate pairs ($a+bi$ and $a-bi$). Fundamental Theorem of Algebra: An $n$ degree polynomial has $n$ complex roots, factoring into real linear and irreducible quadratic factors.

Mathematical induction proves a statement $P_n$ is true for all integers $$n. less a$$.

Proof by contradiction assumes the negation of a statement is true. This assumption is used to derive a mathematical impossibility (a contradiction).

Systems of linear equations can be classified as consistent (at least one solution) or inconsistent (no solution). They can be underspecified ($m < n$) or overspecified ($m > n$).