You Gotta Know These Computation Areas

This You Gotta Know article is devoted to twelve computational areas that will help the most in solving the sorts of math questions that come up in NAQT invitational series. NAQT's collegiate sets tend to have very little computation and much of that is in the context of a specific field (physics, economics, chemistry, etc.).

  1. Pythagorean Triples. Almost certainly, the most important things to know are the basic sets of integers that satisfy the Pythagorean Theorem (a2 + b2 = c2) and could be the side lengths of a right triangle. These are called Pythagorean Triples and the simplest ones are 3-4-5, 5-12-13, 7-24-25, and 8-15-17. Note that any multiple of a Pythagorean Triple is also a Pythagorean Triple so that 6-8-10, 15-20-25, and 300-400-500 are also ones by virtue of 3-4-5 being one.

  2. Matrices. Every team should be able to add, subtract, multiply, take the determinant of, transpose, and invert matrices, particularly two-by-two ones.

  3. Vectors. Every team should be able to find the length of a vector, and add, subtract, find the angle between, the dot product of, and the cross product of two vectors.

  4. Solids. Teams should be able to calculate the volume and surface area of simple geometric figures including the sphere, cone, cylinder, pyramid, hemisphere, prism, and parallelepiped.

  5. Plane Figures. Teams should be able to calculate the areas of triangles (especially equilateral triangles), trapezoids, parallelograms, rhombi, and circles using different angles and lengths.

  6. Similar Figures. The areas of similar figures are related by the square of any corresponding length and the volumes are related by the cube of any corresponding length. For instance, if a square has a diagonal that is 30% longer than another square, it has an area that is (1.30 x 1.30 = ) 1.69 times as great (69% greater). Similar reasoning applies to perimeters, side lengths, diameters, and so forth.

  7. Permutations. Teams should be able to compute the number of permutations and combinations of n objects taken m at a time. They should also have memorized the first eight (or so) values of the factorial function to make this easier.

  8. Logarithms. Teams should be familiar with basic operations of logarithmic math: simplifying the logarithm of a product, difference, or power, and converting from one base to another.

  9. Complex Math. Teams should be familiar with the symbol i representing the imaginary square root of -1, basic operations on complex numbers, graphing complex numbers, and converting complex numbers to magnitude-angle form.

  10. Divisibility Rules. Teams should be able to quickly apply the divisibility rules for small integers (2 through 11) to large integers.

  11. Polynomial Math. Teams should be able to quickly add, subtract, multiply, divide, factor, and find the roots of low-degree polynomials.

  12. Calculus. Teams should be able to find the derivative, integral, slope-at-a-point, local extrema, and critical points of polynomial, trigonometric, and other common functions.

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