六至八年级Computation计划中

Programming for mathematics

Writing short Python programs that use variables, loops, conditionals and functions to explore mathematics: generating sequences, testing conjectures on many cases, and representing numbers in binary.

math.comp.programming-intro

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前置知识

Repetition and decisions Variables and algebraic expressions

9 项必需的前置知识（一直追溯到底）

你将学会

Write programs with variables, loops, conditionals and functions to compute sums, tables and sequences.
Use a program to test a conjecture on many cases, and explain why this is evidence but not proof.
Convert between decimal and binary representations, by hand and in code.

定义

A program variable names a stored value that can change while the program runs.
A program function takes inputs (arguments) and returns an output, possibly with other effects.
In base 2, the digits 0 and 1 count ones, twos, fours, eights and so on.

定理

Every positive whole number has exactly one binary representation without leading zeros.
Checking a statement for the first million cases is experimental evidence, not a proof.
Floating-point numbers are approximations: in Python, 0.1 + 0.2 == 0.3 evaluates to False.

常见误解

Reading '=' in a program (assignment) as mathematical equality.
Believing a program that runs without errors must compute the right answer.
Assuming computer arithmetic with decimals is exact.

通向

Algorithms, correctness and efficiency First steps in Lean Numerical methods The λ-calculus

可选发现

Numerical analysis Floating-point rounding, visible in 0.1 + 0.2, is studied systematically in numerical analysis.
Experimental mathematics: conjectures and counterexamples Computer experiments suggest conjectures and find counterexamples in research mathematics.
← Functions

学习路径

Computer science and formal methods