Numbers
Natural numbers
What natural numbers are, how the set is written and what properties it has. Division with remainders and the question of whether zero is a natural number.
Integers
What integers are, how the set Z is written and what properties it has. Even and odd numbers, division with remainders and closure under the basic operations.
Rational numbers
Rational numbers are all the numbers that can be written as a fraction. Definition, properties, recurring decimals, converting recurring decimals to fractions and the puzzle of 0.999…
Why 0.999… equals 1
Why 0.999… with infinitely many nines is exactly equal to one. Proofs using thirds, multiplying by ten, the difference and the sum of a geometric series, and why we find it hard to believe.
Irrational numbers
Irrational numbers cannot be written as a fraction and have an infinite non-recurring decimal expansion. Definition, properties and open questions about pi and e.
Real numbers
Real numbers are formed by the union of the rational and irrational numbers. The notation for the set R, its properties and the idea of the real number line.
Algebraic numbers
An algebraic number is a root of a polynomial with integer coefficients. Explanation through examples, the precise definition and links to further sources.
Complex numbers
Complex numbers step in where the real numbers fall short, for example with square roots of negative numbers. Cartesian form, the imaginary unit and operations with them.
Where Euler’s number comes from
Where Euler’s number e comes from. An explanation through compound interest and continuous compounding: why £1 at 100% interest grows to about £2.718.
Divisibility
A number is divisible by 2 if its last digit is even. The divisibility test explained with examples, including negative numbers.
Prime numbers
What prime numbers are and what properties they have. The definition, Fermat’s little theorem and the unsolved questions about primes.
The fundamental theorem of arithmetic
The fundamental theorem of arithmetic: every positive integer greater than 1 can be written uniquely as a product of primes. Proofs of existence and uniqueness.
Converting between number systems
How to convert numbers between number systems. The method from base 10 to base 2 and back, why it works and other bases.
Sets
What a set is and how to write one. Order and repeated elements, size and equality of sets, subsets, the empty set and the power set.
Interest
What interest is and how it differs from the interest rate. Working out simple and compound interest, explained with clear examples.
Interval
What an interval is and how to work with it. Open and closed intervals, notation with brackets, an interval as a set and its geometric interpretation.
Infinity
What infinity is and why we cannot picture it. The sideways figure eight symbol, rules for calculating with infinity and how it resembles zero.
Completing the square
Completing the square turns a quadratic function into the form (x+m)^2+n. A step-by-step algorithm, including the case where a is not equal to 1.
Least common multiple
What the least common multiple is and how to find it for two or more numbers. Explained by listing multiples, with examples.
Greatest common divisor
What the greatest common divisor is and how to work it out. The Euclidean algorithm explained step by step, with examples.
Properties of operations
An operation is commutative if the order of the operands does not matter. Examples with numbers, set operations, vectors and logic, and when commutativity fails.
