Chapter 1 ๐ข
Patterns in Mathematics
Quick, colourful revision notes โ easy to read before your test!
Chapter Contents
1.1 What is Mathematics?
Patterns are everywhere around us โ in nature, at home, in school, in the motion of the sun, moon and stars, in shopping, cooking, games and technology.
Maths is both an art (creative, fun to explore) and a science (needs proof and explanation).
Patterns in genomes โ helped in diagnosing and curing diseases.
1.2 Patterns in Numbers
The most basic patterns in maths are patterns of whole numbers: 0, 1, 2, 3, 4, โฆ
Number sequence โ a list of numbers that follow a fixed rule.
| Sequence Name | Numbers |
|---|---|
| All 1's | 1, 1, 1, 1, 1, โฆ |
| Counting numbers | 1, 2, 3, 4, 5, โฆ |
| Odd numbers | 1, 3, 5, 7, 9, โฆ |
| Even numbers | 2, 4, 6, 8, 10, โฆ |
| Triangular numbers | 1, 3, 6, 10, 15, โฆ |
| Squares | 1, 4, 9, 16, 25, โฆ |
| Cubes | 1, 8, 27, 64, 125, โฆ |
| Virahฤnka numbers | 1, 2, 3, 5, 8, 13, โฆ |
| Powers of 2 | 1, 2, 4, 8, 16, 32, โฆ |
| Powers of 3 | 1, 3, 9, 27, 81, โฆ |
1.3 Visualising Number Sequences
Pictures make number patterns easy to understand. Here's how some sequences look when drawn as dots:
Fig 1: Triangular numbers and Square numbers made of dots
1.4 Relations among Number Sequences
Number sequences can be related to each other in surprising ways!
1 = 1 | 1+3 = 4 | 1+3+5 = 9 | 1+3+5+7 = 16 | 1+3+5+7+9+11 = 36
Fig 2: Each coloured "L" (called a gnomon) is one odd number โ together they build a bigger and bigger square!
Because this picture can be made bigger and bigger for any size square, this pattern goes on forever.
Sum of first 100 odd numbers = 100ยฒ = 10,000
Adding counting numbers up and down also gives square numbers:
1.5 Patterns in Shapes
Regular polygons โ shapes with equal-length sides AND equal angles:
Fig 3: Regular Polygons โ sequence continues: Nonagon (9), Decagon (10) โฆ
Complete Graphs โ every point joined to every other point by a line:
Fig 4: Complete Graphs K2 to K6
Stacked Squares and Stacked Triangles are shapes built from small squares/triangles (see Fig 1 above โ they follow the square and triangular number patterns).
Koch Snowflake โ replace every straight line "โ" with a "speed bump" shape, again and again:
Fig 5: Koch Snowflake โ each round makes tinier and tinier "bumps". Segments: 3, 12, 48, 192 โฆ (3 ร Powers of 4)
1.6 Relation to Number Sequences
Shape sequences often connect beautifully with number sequences!
| Shape Sequence | Related Number Sequence |
|---|---|
| Regular Polygons โ sides | 3, 4, 5, 6, 7, 8, 9, 10 โฆ (counting numbers from 3) |
| Regular Polygons โ corners | Same as number of sides! |
| Complete Graphs โ lines | 1, 3, 6, 10, 15 โฆ (triangular numbers) |
| Stacked Squares | 1, 4, 9, 16, 25 โฆ (square numbers) |
| Stacked Triangles | 1, 4, 9, 16, 25 โฆ (square numbers) |
| Koch Snowflake โ segments | 3, 12, 48, 192 โฆ (3 ร powers of 4) |
๐ฏ Quick Summary
- Mathematics = search for patterns + explanations of why they exist.
- Key number sequences: counting, odd, even, triangular, square, cube, Virahฤnka, powers of 2 & 3.
- Adding odd numbers from 1 โ always gives square numbers.
- Adding counting numbers up-and-down โ also gives square numbers.
- Geometry studies shape patterns: regular polygons, complete graphs, stacked shapes, Koch snowflake.
- Shape sequences often match number sequences (e.g. polygon sides = counting numbers).
โ๏ธ Notes by @edugrown โ happy revising! ๐