Real Numbers & Order of Operations

Before manipulating unknowns, you must operate fluently on numbers. This week refreshes the real-number system, sign rules, and the universal evaluation order (PEMDAS).

The number system in layers

• Naturals (ℕ): 1, 2, 3, …
• Whole numbers: 0, 1, 2, …
• Integers (ℤ): …, −2, −1, 0, 1, 2, …
• Rationals (ℚ): any number expressible as a/b with a, b ∈ ℤ, b ≠ 0. Includes terminating and repeating decimals.
• Irrationals: numbers like π, e, √2 — not expressible as a fraction.
• Reals (ℝ): rationals ∪ irrationals.

Rationals are the only set among these that's closed under all four arithmetic operations (division excluding zero). Naturals aren't closed under subtraction; integers aren't closed under division.

PEMDAS / order of operations

1. Parentheses
2. Exponents
3. Multiplication / Division (left to right)
4. Addition / Subtraction (left to right)

Example: 8 − 2·3² evaluates as 8 − 2·9 = 8 − 18 = −10. The exponent comes before multiplication; multiplication before subtraction.

Sign rules

• Two negatives multiplied or divided yield a positive: (−3)(−4) = 12.
• Subtracting a negative is the same as adding: 7 − (−2) = 9.
• Distribution carries the negative across all terms: −(x − 6) = −x + 6, not −x − 6.

Absolute value

The absolute value of a real number x, written |x|, is its distance from zero on the number line. Distance is never negative, so |x| ≥ 0 always.

Piecewise definition:

$|x|=\{\begin{array}{ll}x& \text{if }x\ge 0\\ -x& \text{if }x<0\end{array}$

So $|5|=5$, $|-7|=7$, $|0|=0$. The bars don't flip a positive number — they strip the sign. A consequence: $\sqrt{x^2}=|x|$, not $x$, because the square root must return a non-negative result.

Distributive property

a(b + c) = ab + ac. The single most-used identity in algebra. Used to expand parentheses, then re-used in reverse to factor. Watch the sign on a: a negative coefficient distributes as a negative to every inner term.

Why this matters

1. Algebraic structure learned. PEMDAS, sign rules, and the distributive property over the real numbers — the grammar of every numerical expression.
2. ML object with the same structure. Loss-function expressions like $\frac{1}{2m}\sum (h(x)-y{)}^2$ are nested arithmetic at scale: parentheses group sums, exponents come before products, and a misplaced sign flips the gradient.
3. Computational payoff. Vectorized array code ((1/(2*m)) * np.sum(...)) evaluates these expressions in parallel across thousands of samples. A precedence slip in the formula cascades into wrong numbers across an entire batch — silent and expensive. PEMDAS discipline on paper transfers directly to bug-free vectorized code.