Number Systems

Every calculation starts with numbers. Learn natural numbers, whole numbers, integers, rationals, irrationals, reals and complex numbers; primes, composites, even/odd; the number line, intervals and absolute value; inequalities and ordering; and how the number-system hierarchy fits together.

Engineers measure lengths (positive reals), count parts (natural numbers), record temperatures below zero (integers), store sensor readings as fractions (rationals), meet √2 in diagonals and π in circles (irrationals), and describe AC voltages with complex numbers.

Knowing which set a quantity belongs to tells you what operations make sense and which errors to watch for — e.g. you cannot take a real square root of a negative number, and computers cannot store most irrationals exactly.

An analogy

Number systems are like nested boxes: natural numbers sit inside integers, integers inside rationals, rationals inside reals, and reals inside the complex numbers — each bigger box lets you solve problems the smaller box can't.

The families

Natural numbers count; whole numbers add 0; integers add negatives; rationals are ratios of integers (terminating or repeating decimals); irrationals (√2, π, e) have non-repeating decimals; reals are all points on the number line; complex numbers add an imaginary axis.

Primes and parity

A prime has exactly two divisors (2, 3, 5, 7, 11…); composites have more; 1 is neither. Even numbers are divisible by 2. Every integer > 1 factors uniquely into primes (fundamental theorem of arithmetic) — the basis of RSA encryption.

Intervals, ordering and tolerance

Intervals describe allowed ranges: a resistor of 100 Ω ± 5 % lies in [95, 105]. |x − a| ≤ r is a tolerance band. Inequalities flip direction when multiplying or dividing by a negative number.

Numbers in computers

Integers are exact within their bit width (overflow beyond); floating-point numbers approximate reals with ≈ 7 (float) or ≈ 16 (double) significant digits. 0.1 cannot be stored exactly in binary — the root of many numerical surprises.

Formulas

• Hierarchy: ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
• Absolute value: |x| = x if x ≥ 0, −x if x < 0
• Absolute-value inequality: |x − a| < r ⇔ a − r < x < a + r
• Imaginary unit: i² = −1

Worked example: Classify numbers

Given: −3, 0, 2/5, √2, 3 + 2i

Find: Which sets each belongs to.

• −3: integer, rational, real, complex
• 0: whole number, integer, rational, real, complex
• 2/5: rational, real, complex
• √2: irrational, real, complex
• 3 + 2i: complex only (not real)

Answer: Each number belongs to every set above its smallest set

Worked example: Tolerance interval

Given: Shaft diameter 25.00 mm ± 0.02 mm

Find: Interval and inequality.

• |d − 25.00| ≤ 0.02
• 24.98 ≤ d ≤ 25.02

Answer: d ∈ [24.98, 25.02] mm

Common mistakes

• Wrong: Thinking 1 is prime. — Right: Primes have exactly two divisors; 1 has one.
• Wrong: Not flipping inequality when dividing by a negative. — Right: −2x < 6 ⇒ x > −3.

Applications

• Tolerances: Intervals and absolute value define acceptable part sizes.
• Digital systems: Integers and fixed/floating-point formats in processors.
• AC circuits: Complex numbers represent impedance and phase.

Key takeaways

• ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ.
• Primes build all integers.
• |x − a| ≤ r describes tolerances.
• Computers approximate reals.

Going further

• Countability: ℚ is countable, ℝ is not (Cantor).
• IEEE 754: Floating-point format and rounding.
• Modular arithmetic: Integers mod n — see cryptographic mathematics.