South 고등학교 2학년 Mathematics and Culture
Chapters: 4
1. Art and mathematics
Maths in music · Maths in visual art · Maths in literature · Maths in film
- Maths in the Arts: Music, Pictures, Tiles, Poems and Film – Art is full of maths. Musical notes are ratios of string length (half length = octave, 2/3 length = fifth). Painters use perspective and the golden ratio. Tiles fit when their corner angles add up to 360°. Poems repeat sound patterns. A film is 24 pictures per second.
2. Life and mathematics
Maths in sport · Maths in games · Maths in digital technology · Maths of voting
- Mathematics in Sport – Maths helps athletes and organisers. A ball thrown at speed v and angle θ travels R = v² sin 2θ ÷ g (best at 45° without air drag). Statistics such as mean, median, batting average and strike rate summarise performance. Scheduling counts matches: a round-robin of n teams has n(n−1)/2 matches, a knockout has n−1.
- Game Theory: Making the Best Choice When Others Choose Too – Game theory studies decisions where your result depends on what others choose. A pay-off matrix lists each player's gain for every pair of choices. A dominated strategy is always worse and can be removed. A Nash equilibrium is a pair of choices where no player gains by changing alone; the prisoner's dilemma shows it can be worse for everyone than cooperating. In a zero-sum game, the play-safe (maximin/minimax) strategies meet at a saddle point when the game is stable; otherwise players use a mixed strategy, found by drawing expected-pay-off lines and taking the highest point of the lower edge.
3. Society and mathematics
Ethnomathematics and architecture · Maths of braille · Data from mass media · Decision-making for value consumption
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
- Data Analysis – Data analysis means turning raw data into answers. It follows a cycle: ask a question, collect data, clean it (remove errors, repeats and blanks), organise and transform it, analyse it with summaries such as mean, median, range and patterns, show it with a good chart, and draw a careful conclusion. Watch for outliers, small samples and bias, and remember that a correlation between two things does not prove that one causes the other. Data must also be stored safely and used with permission.
- Decision Making: How to Choose Well, Step by Step – Decision making means choosing the best option from two or more choices. A good decision follows steps: spot the problem, list options, set criteria, give each criterion a weight, score the options, choose, act and then review. A decision matrix turns this into simple numbers.
4. Environment and mathematics
Food and diet data · Air pollution data · Desertification data · Biodiversity data
- Data Analysis – Data analysis means turning raw data into answers. It follows a cycle: ask a question, collect data, clean it (remove errors, repeats and blanks), organise and transform it, analyse it with summaries such as mean, median, range and patterns, show it with a good chart, and draw a careful conclusion. Watch for outliers, small samples and bias, and remember that a correlation between two things does not prove that one causes the other. Data must also be stored safely and used with permission.