Book of Proof, third edition
Richard Hammack · Free textbook
Read §1.9 on number systems; this course explicitly includes zero in ℕ.
Open the sourceWhy enlarge the number system?
New numbers repair operations that used to get stuck.
Prerequisites: 01 · What is a number?
Understand
Objectives
- Identify an operation or equation motivating each number-system extension.
- Decide whether an integer pair represents a rational.
- Check equivalence of two fraction names by cross-products.
Start with something you can see
Start with three counters. Predict which number names the result of removing five: no collection of counters represents a debt of two. A signed balance does. Dividing one counter among three people requires thirds. A square with side one has a diagonal whose square is two, but no fraction of integers has that value. Each extension answers a particular missing-operation or missing-measurement question.
Give the idea a precise name
In this course ℕ={0,1,2,…} is the natural numbers including zero; ℤ adds negative integers. ℚ contains ratios of integers, ℝ includes measurements such as √2, and ℂ includes numbers a+bi for real a,b and i²=−1. A rational representation (a,b) requires integers a and b with b≠0. Two such pairs (a,b) and (c,d) represent the same rational number precisely when ad=bc.
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ
The containment symbol means every element of the system on its left is also in the system on its right. It does not mean the systems have identical operations or order: the usual real order does not extend to an ordered-field order on ℂ. In particular (√2,1) is not an allowed integer pair for this construction of ℚ, and (1,0) has an invalid denominator.
Work one small world
The valid pairs (1,2) and (3,6) name the same rational because 1×6=2×3. Their sum with 1/3 is 1/2+1/3=3/6+2/6=5/6. For complex numbers, i(x+iy)=−y+ix: multiplying by i turns the plane through a quarter-turn. The longer creator lecture discusses this geometric multiplication at 20:43. [1]
| Question | First named solution | System |
|---|---|---|
| 2−5 | −3 | ℤ |
| 1÷3 | 1/3 | ℚ |
| x²=2, x≥0 | √2 | ℝ |
| x²=−1 | i | ℂ |
Counterexample to “any numerical pair defines a rational”: (√2,1) has a noninteger first entry and (1,0) has denominator zero. Neither is an allowed pair in this construction, even though (3,4) is. Writing a pair is not enough: its domain conditions matter.
THE BRIDGEA change in notation can preserve a value, while an extension supplies genuinely new values or operations.
WHERE THIS IDEA STOPSThe convention 0∈ℕ is declared here; other books use a different convention. Not every equation has a solution in every system.
Experiment
Use the table to predict the smallest named system for each question. For fractions, compare (1,2) and (3,6) with cross-products before computing their sum.
Which equation forces us beyond positive whole numbers, and which forces us beyond fractions?
Check & explain
Which pair is allowed in the integer-pair construction of a rational?
- (√2,1)
- (1,0)
- (3,4)
What is 1/2+1/3 as a fraction with denominator six? Enter its numerator.
Teach back
Give one equation without a rational solution but with a real solution; then explain why a pair with denominator zero is not a fraction.
- Name the new system and the equation.
- State that rational pairs require integer entries.
- Exclude a zero denominator rather than silently dividing by zero.
Read deeper
Algebra and Trigonometry 2e
OpenStax · Open textbook
Read OpenStax Algebra and Trigonometry 2e §§1.1 and 2.4 for real and complex arithmetic.
Open the sourceComplex number fundamentals | Ep. 3 Lockdown live math
3Blue1Brown · Lecture · 82 min
Begin the 82-minute complex-number lecture at 20:43 for multiplication by i.
Open the sourceLecture 3: Cantor's Remarkable Theorem and the Rationals' Lack of the Least Upper Bound Property
MIT OpenCourseWare · Course lecture · MIT OCW
Official OCW lecture 3: √2 and rational incompleteness; omit the preceding historical anecdote.
Open the source