Ultimate Guide: Mathematics: Analysis and Approaches (IB)

Standard form

A way of writing numbers as ax10^k where 1≤a≤10 and k is an integer.

Laws of indices

Rules that simplify expressions involving exponents, including product, quotient, power, and zero exponent laws.

Partial fractions

A method to simplify rational expressions into the sum of two or more fractions with constant numerators.

Logarithm

The inverse of an exponent; if a^x = b, then ext{log}_a b = x.

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant, called the common difference.

Geometric sequence

A sequence in which there is a constant ratio between consecutive terms, known as the common ratio.

Sum to infinity

The sum of an infinite geometric sequence is given by S∞ = \frac{u1}{1 - r} when |r| < 1.

Proof by contradiction

A method of proving a statement by assuming the opposite is true, leading to a contradiction.

Binomial Theorem

A formula for expanding expressions of the form (a+b)^n using binomial coefficients.

Permutation

The arrangement of objects in which the order matters.

Combination

The selection of objects where the order does not matter.

Argand diagram

A graphical representation of complex numbers where the x-axis represents the real part and the y-axis represents the imaginary part.

Cartesian form

Notation for complex numbers expressed as z = a + bi where a and b are real numbers.

Modulus of a complex number

The distance of the complex number from the origin, calculated as |z| = \sqrt{a^2 + b^2}.

Exponentials

Functions in the form f(x) = a(b^x) where a is a constant and b is the base.

Logarithmic function

The inverse function of an exponential function, given as f(x) = \text{log}_b (x).

Linear function

A function where the highest degree of the variable is one, expressed as f(x) = mx + b.

Domain

The set of all possible input values for a function.

Range

The set of all possible output values produced by the function.

Inverse function

A function that reverses the effect of the original function; if f(x) = y, then f^{-1}(y) = x.

Even function

A function that satisfies f(-x) = f(x) for all x in its domain.

Odd function

A function that satisfies f(-x) = -f(x) for all x in its domain.

Critical point

A point on a function where its derivative is zero or undefined, indicating potential maxima or minima.

Null Hypothesis (H0)

The hypothesis stating there is no effect or no difference, tested using sample data.

Alternative Hypothesis (H1)

The hypothesis suggesting a potential effect or difference, which is believed to be true if the null hypothesis is rejected.

Statistics

A branch of mathematics dealing with data collection, analysis, interpretation, and presentation.

Mean

The arithmetic average of a set of numbers, calculated by adding all values and dividing by the number of observations.

Median

The middle value in a set of data when arranged in order.

Mode

The value that occurs most frequently in a data set.

Standard deviation (σ)

A measure of the dispersion of data points around the mean.

Confidence interval

A range of values that is likely to contain the population parameter with a certain level of confidence.

Bayes' theorem

A mathematical formula for updating probabilities based on new evidence.