Variables and algebraic expressions

前置知识

6 项必需的前置知识（一直追溯到底）

你将学会

• Write algebraic expressions for situations and evaluate them for given values, including negative ones.
• Simplify expressions by combining like terms, and expand and factorise using the distributive law.
• Use the laws of exponents aᵐaⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ for whole-number exponents.

定义

• A variable is a letter that stands for any number from a given set.
• Two expressions are equivalent if they have the same value for every value of their variables.
• For a whole number n ≥ 1, aⁿ is the product of n factors each equal to a; a⁰ = 1 for a ≠ 0.

定理

• a(b + c) = ab + ac for all numbers a, b and c.
• aᵐ · aⁿ = aᵐ⁺ⁿ and (ab)ⁿ = aⁿbⁿ.
• Testing an equivalence with a few values can refute it but cannot prove it.

常见误解

• Writing (a + b)² = a² + b².
• Treating 2x and x² as the same.
• Believing x must be positive, or that different letters must have different values.

通向

可选发现

• Rings, ideals and fields The rules for manipulating expressions are the laws of a commutative ring, so the same manipulations are valid wherever those laws hold, for example with remainders modulo n.

学习路径

• Adult beginner pathway

延伸阅读

• I. M. Gelfand and A. Shen, Algebra (1993)