This opening chapter presents mathematics as the search for patterns and the explanations of why those patterns exist. It introduces number sequences as the most basic patterns of whole numbers \(0, 1, 2, 3, \dots\) studied in number theory. Key sequences include counting numbers, odd numbers, even numbers, triangular numbers \(1, 3, 6, 10, 15, \dots\), square numbers \(1, 4, 9, 16, 25, \dots\), cubes \(1, 8, 27, 64, \dots\), Virah\u0101nka (Fibonacci-like) numbers, and powers of 2 and 3. Many sequences can be visualised with dot pictures, and the chapter shows beautiful relations: the sum of the first \(n\) odd numbers equals \(n^2\) (so \(1+3+5+7+9+11=36\)), adding counting numbers up and down also gives square numbers, and adding consecutive triangular numbers gives squares. It then turns to patterns of shapes \u2014 regular polygons, complete graphs, stacked triangles and squares, and the Koch snowflake \u2014 studied in geometry, and links these shape sequences back to number sequences (sides of regular polygons give counting numbers from 3, lines in complete graphs give triangular numbers).
What is Mathematics?Patterns in NumbersVisualising Number SequencesRelations among Number SequencesPatterns in ShapesRelation to Number Sequences
Key terms
Number sequence
An ordered list of numbers formed by a fixed rule, such as odd numbers or square numbers.
Triangular numbers
The sequence \(1, 3, 6, 10, 15, \dots\) formed by dots that can be arranged in a triangle.
Square numbers
The sequence \(1, 4, 9, 16, 25, \dots\); the number \(n^2\) of dots arranged in a square.
Cubes
The sequence \(1, 8, 27, 64, 125, \dots\), i.e. \(n^3\) dots arranged in a cube.
Regular polygon
A closed shape with all sides equal and all angles equal, such as an equilateral triangle or square.
Koch snowflake
A shape sequence with \(3, 12, 48, \dots\) line segments (3 times powers of 4), formed by repeatedly replacing each segment with a bump.
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Which branch of mathematics studies patterns in whole numbers?
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