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5 most critical mathematical concepts
Posted by George Richardson on 12/2/2010
In Reply To:5 most critical mathematical concepts Posted by Tim Joy on 12/2/2010
Here's a proposed list of the 10 greatest mathematical concepts, designed to aim at the purpose of Tim's question.
I hope you enjoy it. I had fun trying to think about it.
...George
Ten most critical/valuable mathematical concepts
1. The positional number system
1. The numerals “3”, “4”, “5” and “6” in 34.56 mean dramatically different things.
2. It’s crucial to be able to compute in this decimal number system -- every calculator and spreadsheet uses it.
3. And the best measuring system in the world (the metric system), used in all but two countries in the industrialized world, uses it extensively, e.g., 1 cm = 0.01 m.
4. Opinion: It’s not so crucial to be able to compute in fractions -- the most common place we encounter them, where decimals won’t do, is in cookbooks. We’d have to know how to read them, but I’m not sure many people have to know how to compute 3/4 + 5/6.
2. Representing a number by a letter: Literal arithmetic.
1. The key to algebra and all mathematics beyond arithmetic and elementary geometry
3. Cartesian coordinate system: Plotting y versus x on a pair of perpendicular numberlines.
1. Within 50 years of Descartes’s development of this idea, Newton and Leibniz built on it to create the basic elements of calculus, including the fundamental theorem of calculus.
4. The concept of accumulation
1. One of the two crucial concepts in calculus (and the world)
2. Represents anything from inventories to electrical capcitance to self-esteem
5. The concept of the rate of change of a varying quantity
1. One of the two crucial concepts in calculus (and the world)
2. Represents anything from velocity of a moving object to population growth to the loss of polar ice
6. The fundamental theorem of calculus
1. ... on probably anybody’s list of the ten greatest ideas in history of humankind.
2. In math-speak: Some definite integrals can be computed by anti-differentiation.
3. In more accessible language, sort of: Some accumulations representable by a function f can be computed from the slope of the graph of y = f(x).
4. Or the simpler “area theorem”: The rate of change of the area under a curve f(x) over a region [a,x] equals the slope of f at x.
5. ... none of which even gives the barest hint of the significance of this discovery.
7. Compounding accumulation
1. Crucial to understanding (and maybe eventually dealing with) debt, population growth, inflation, global warming and cooling
8. The idea of mathematical objects as “models”
1. “Number” as a model
1. “7” is the one thing that all groups of 7 things have in common -- it’s a model of “seven-ness.”
2. “Geometric objects” as models
1. A line is the idea in common to all “straight things” -- a model of uni-dimensional straightness.
2. A sphere is the idea in common to all balls -- a model of three-dimensional roundness.
3. Area under a curve as a model of
1. Population, when the curve represents the population net growth rate,
2. Pollution, when the curve represents the pollution net growth rate,
3. Self-esteem, when the curve represents the net growth rate of self-esteem.
4. The slope of a curve as a model of
1. Rate of change of population, when the curve represents the size of the population
2. Area, when the curve represents volume
5. The great leverage point idea for helping young people realize that thoughts are models, not “reality”, and that as models they can be reflected upon, discussed, debated, evaluated, proved wrong, changed, and improved
9. Multiplication
1. As repeated addition
2. As a more general model of “replication,” e.g., Progress = Effort * Effectiveness, where Progress is measured in accomplishments per week, Effort is in hours per week, and Effectiveness is in accomplishments per hour
10. Comparison
1. Ratios
2. Differences
3. All influences in the real world are best thought of not in absolute terms but in comparative terms:
1. Influence of food supply on population health is best thought not as food itself but food per capita
2. Influence of atmospheric CO2 concentration on global warming is best thought of not as the accumulation of atmospheric CO2 itself, but the comparson of that to what has been typical in the last 100 years, or 1,000 years, or 1,000,000 years.
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