The Timpson Lyceum · Geometry (2024)

Comparing and Measuring Segments – Lecture 6 (Geometry)

Comparing and Measuring Segments – Lecture 6

Overview

This lecture turns the qualitative idea of congruence into a quantitative one: it shows how to assign a length to a segment and why every segment has a well-defined length. Three big ideas organize the chapter: a length is a number together with a unit; once you fix a unit, every segment has a length in that unit; and conversely, for any positive real number there is a segment of that length. Getting there requires a short tour of the number systems (naturals, integers, rationals, reals) and the fact that every real number has a decimal expansion. Two new postulates — the Archimedean postulate and the Completeness postulate — make the measuring process rigorous, and a handful of theorems tie length back to the congruence relation studied earlier.

Key Concepts

Recap of congruence. Congruence is an undefined term whose meaning comes from postulates. It is a two-term relation (a predicate that returns true/false for a pair of segments). The key facts carried in from the previous chapter:

• Segment construction: given a segment $AB$, a point $A'$ on a line, and a side of $A'$, there is a unique point $B'$ on that side with $\overline{A'B'} \cong \overline{AB}$.
• Equivalence relation: congruence is reflexive ($a \cong a$), symmetric ($a \cong b \Rightarrow b \cong a$), and transitive ($a \cong b$ and $b \cong c \Rightarrow a \cong c$).
• Additivity/subtractivity: equals added to equals are equal, and equals subtracted from equals are equal (for adjacent congruent sub-segments).
• Order of segments: $AB < A'B'$ means there is a point $B''$ with $\overline{AB} \cong \overline{A'B''}$ and $B''$ between $A'$ and $B'$; $AB > A'B'$ means that same $B''$ lies beyond $B'$ (so $B'$ is between $A'$ and $B''$).

The three big ideas of this chapter.

• Length is a number with a unit. Saying "17" is meaningless; "17 meters" is a length. A length is a real number paired with a chosen unit.
• Fix a unit ⇒ every segment has a length. Any unit (foot, meter, mile, light-year) works; changing the unit only changes the number, not the existence of a length.
• Fix a length ⇒ a segment exists. For any real number you name, a segment of exactly that length can be constructed from a starting point.

Number systems. Each system fills a gap left by the previous one:

• Naturals $\mathbb{N}=\{0,1,2,\dots\}$ — closed under addition, not subtraction.
• Integers $\mathbb{Z}$ — add negatives so subtraction always works.
• Rationals $\mathbb{Q}$ — fractions $p/q$ (integer over nonzero integer); close division.
• Reals $\mathbb{R}$ — add the irrationals (e.g. $\sqrt{2}$, $\pi$, $\sqrt{3}$, $\tfrac{\sqrt{2}}{2}$), numbers that cannot be written as a fraction.

Repeating decimals and rationals coincide. Every real number has a decimal expansion. A number is rational if and only if its decimal expansion eventually repeats (terminating decimals count — they end in repeating $0$s). Irrationals are exactly the reals whose decimals never settle into a repeating pattern. Doing long division always forces a remainder to recur, so every rational repeats; conversely the "multiply, shift, subtract" trick converts any repeating decimal back into a fraction. The reals are precisely all numbers with a decimal expansion.

Inequalities on the reals (distinct from the segment order, despite the shared symbols):

• $x > y$ means $x - y$ is positive; $x < y$ means $x - y$ is negative.
• $x$ is positive $\iff x > 0$; $x$ is negative $\iff x < 0$.
• Transitivity: $x > y$ and $y > z \Rightarrow x > z$ (since the sum of two positives is positive).
• Mixed addition: if $x > y$ and $u \ge v$, then $x + u > y + v$.

The measuring process. To measure $\overline{AB}$: pick a unit — any reference segment (a meter is the standard rod in France; a foot was once the king's foot). Lay copies of the unit end to end until you land on $B$ or overshoot; that whole-number count is the integer part. Then refine with a smaller sub-unit (a tenth of the unit is natural but any subdivision works), repeating to get successive decimal digits. The result is the segment's decimal expansion in that unit. Some lengths never terminate: the diagonal of a unit square is $\sqrt{2}$ (by the Pythagorean theorem), an infinite non-repeating expansion — Aristotle called such pairs incommensurable.

Postulate — Archimedean. If $\overline{AB}$ lies on line $L$ and $\overline{CD}$ is any unit segment, then there exist a natural number $n$ and points $A=A_0, A_1, A_2, \dots, A_n$ on $L$ with $\overline{CD} \cong \overline{A_0A_1} \cong \overline{A_1A_2} \cong \cdots \cong \overline{A_{n-1}A_n},$ and either $A_n = B$ (the unit fits exactly) or $B$ lies between $A_{n-1}$ and $A_n$ (you stop one step past $B$). This guarantees the second big idea: any segment can be measured by repeatedly laying down a unit.

Postulate — Completeness. For any real number $x$ and any point $A$ on a line $L$, there exists a point $B$ on $L$ such that $\overline{AB} = x$ (in the chosen units). This is the third big idea: every real length is realized by some segment.

Notation. $\overline{AB}$ with no further mark denotes the segment (a geometric figure); written as a length $\overline{AB}$ it denotes the number that is its measure.

Length theorems.

• Theorem 5.1: if $\overline{AB} \cong \overline{A'B'}$ then $\overline{AB} = \overline{A'B'}$ (congruent segments have equal length).
• Theorem 5.2 (converse): if $\overline{AB} = \overline{A'B'}$ then $\overline{AB} \cong \overline{A'B'}$.
• Theorem 5.3: $\overline{AB} > \overline{CD}$ (segment order) $\iff \overline{AB} > \overline{CD}$ (number order).
• Unit conversion: if $\overline{AB} = x$ in $CD$-units and $\overline{CD} = y$ in $EF$-units, then $\overline{AB} = xy$ in $EF$-units — exactly the reasoning behind "$7$ ft $\times\ 12 = 84$ in."

Worked Example

Goal: show that the repeating decimal $0.\overline{9}$ equals $1$, i.e. that this repeating decimal has a rational (in fact integer) value. The method — set the number equal to $x$, shift the decimal by multiplying by a power of $10$, subtract to cancel the repeating tail, then solve — is the general recipe for converting any repeating decimal to a fraction.

The same procedure handles an arbitrary repeating block. For $x = 137.\overline{2164}$, the block has length $4$, so multiply by $10^4$ to align the tails:

Every repeating decimal collapses to a fraction this way, confirming the one-to-one correspondence between rationals and repeating decimals.

Summary

• A length is a real number paired with a chosen unit; the number alone says nothing.
• The number systems grow by closure: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$, with the irrationals (e.g. $\sqrt{2}$, $\pi$) completing the reals.
• A number is rational if and only if its decimal expansion eventually repeats; the reals are exactly the numbers with a decimal expansion.
• The Archimedean postulate lets you measure any segment by laying a unit end to end, yielding a (possibly infinite) decimal expansion.
• The Completeness postulate guarantees that every real number is the length of some segment.
• Length and congruence agree: congruent segments are equal in length and vice versa, segment order matches numeric order, and lengths convert between units by multiplication.