Chapter 1 — Foundations of arithmetic and elementary algebra¶
This chapter builds a compact foundation for numerical and algebraic expressions. The goal is to prepare the progression used in later volumes: first we compute with numbers, then we model relationships with functions, and finally we organize linear structures.
1.1 Arithmetic expressions and order of operations¶
An expression such as \(3 + 4 \cdot 2\) is not evaluated naively from left to right. We apply precedence rules: parentheses, powers, multiplication/division, and then addition/subtraction.
Quick examples:
- \(3 + 4 \cdot 2 = 11\);
- \((3 + 4) \cdot 2 = 14\);
- \(2^3 + 5 = 13\).
A simple summary table is useful.
| Level | Operations | Example |
|---|---|---|
| 1 | Parentheses | \((2+1)^2\) |
| 2 | Powers | \(2^4\) |
| 3 | Multiplication and division | \(12/3\), \(4\cdot 5\) |
| 4 | Addition and subtraction | \(8-2+1\) |
1.2 Basic algebraic manipulation¶
In algebra, symbols represent values that may vary. For instance, in \(2x + 3x\), the terms are like terms and can be combined:
We also use the distributive property:
Applying this idea to a concrete case:
Worked example (step-by-step derivation)¶
Simplify the expression \(2(3x-1) - (x+4)\).
1.3 A small algorithm to evaluate linear expressions¶
When an expression has the form \(ax+b\), we can compute its value for specific inputs.
algorithm evaluate_linear_expression(a, b, x)
y <- a*x + b
return y
end
Mental check: if \(a=3\), \(b=-1\), and \(x=4\), then \(y=11\).
1.4 Bridge to Volume 2¶
The linear form \(y=ax+b\) will reappear in Volume 2 as an affine function connected to slope (rate of change) and intercept. At this stage, the key point is that elementary algebra gives a systematic language for reasoning with variables.