๐ข Module 1.2: Number Sets
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1 ๐ข Module 1.2: Number Sets
- Explore the fundamental number sets for calculus;
- Learn their main properties;
- Present characteristic examples and exercises.
1.1 โจ What Are Number Sets?
Number sets are groups of numbers that share common characteristics. The main ones we will use are:
Natural Numbers \((\mathbb{N})\): \(\{0, 1, 2, 3, \ldots\}\)
Used to count objects; they are the non-negative integers.Integers \((\mathbb{Z})\): \(\{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}\)
Include all naturals, their negatives, and zero.Rational Numbers \((\mathbb{Q})\): numbers that can be written as a fraction
\[ \mathbb{Q}=\left\{\frac{a}{b}\;\middle|\;a,b\in\mathbb{Z},\; b\neq 0\right\}. \]
Examples: \(\tfrac{1}{2}\), \(-\tfrac{3}{4}\), \(5=\tfrac{5}{1}\).Irrational Numbers \((\mathbb{R}\setminus\mathbb{Q})\): cannot be expressed as a fraction of integers; have infinite non-repeating decimals (e.g., \(\sqrt{2}\), \(\pi\), \(e\)).
Real Numbers \((\mathbb{R})\): all points on the real line; include rationals and irrationals.
Credit: Phrood (Public domain).
1.2 ๐ง Organization of Number Sets
The sets are organized in a chain of inclusions: \[ \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}. \]
Credit: Mortalmoth (Public domain).
This means, for example, that every natural number is an integer, rational, and real โ but the reverse is not always true.
1.3 ๐ ๐ง Learn More: What is an irrational number?
๐ข An irrational is a real number that cannot be written as \(\dfrac{a}{b}\) with \(a,b\in\mathbb{Z}\) and \(b\neq 0\).
Its decimal expansion is infinite and non-repeating (e.g., \(\pi\), \(e\), \(\sqrt{2}\)).
They naturally appear in: the diagonal of a square (\(\sqrt{2}\)), circumference (\(\pi\)), exponential growth/logarithms (\(e\)).
Historically, the irrationality of \(\sqrt{2}\) shook the Pythagorean school.
1.4 ๐ The Pythagorean School and the โscandalโ of irrationals
Who were they? A philosopherโmathematician community (6th century BC) who claimed that โeverything is numberโ (rational).
The crisis. The hypotenuse of an isosceles right triangle with legs of length 1 has length \(\sqrt{2}\), which is not rational.
The proof of the irrationality of \(\sqrt{2}\) is attributed to Hippasus.
Impact. The first blow to this worldview; it marked the beginning of the formal acceptance of irrationals, essential for the completeness of \(\mathbb{R}\).
1.5 ๐ ๐ง Exercises โ Identifying Irrationals
- Classify as rational or irrational:
- \(\sqrt{9}\) ยท b) \(\sqrt{5}\) ยท c) \(\tfrac{4}{7}\) ยท d) \(\pi\) ยท e) \(0.101001000100001\ldots\)
- Which one is irrational?
- \(\tfrac{7}{3}\) ยท B) \(1.333\ldots\) ยท C) \(\sqrt{2}\) ยท D) \(0.5\)
- True/False:
- Every infinite decimal is irrational.
- \(\sqrt{25}\) is irrational.
- There are more irrationals than rationals.
- Every infinite decimal is irrational.
1.6 ๐ Answer Key (identification)
- Rational (\(=3\)) ยท b) Irrational ยท c) Rational ยท d) Irrational ยท e) Irrational (infinite non-repeating)
- Rational (\(=3\)) ยท b) Irrational ยท c) Rational ยท d) Irrational ยท e) Irrational (infinite non-repeating)
- C (\(\sqrt{2}\))
- F (only non-repeating infinite decimals are irrational) ยท b) F (\(\sqrt{25}=5\) is rational) ยท c) T (reals/irrationals are uncountable, rationals are countable โ Cantor).
1.7 ๐ Historical Note: Georg Cantor (1845โ1918)
Founder of set theory and the modern treatment of infinity.
He proved that \(\mathbb{R}\) (and therefore the irrationals) is uncountable, while \(\mathbb{Q}\) is countable โ there are โmoreโ irrationals than rationals.
1.8 โจ Decimal Representation of Real Numbers
- Rationals: decimal finite or infinite repeating.
- Irrationals: decimal infinite non-repeating.
Quick examples: - \(\frac{3}{4}=0.75\) (finite)
- \(\frac{1}{3}=0.\overline{3}\) (infinite repeating)
- \(\sqrt{2}\approx 1.4142135\ldots\) (infinite non-repeating)
1.9 โจ Repeating Decimals and Generating Fraction
Ex. 1 โ \(\tfrac{1}{3}\):
\(1\div 3 = 0.333\ldots = 0.\overline{3}\) โ rational.
Ex. 2 โ \(\tfrac{4}{11}\):
\(4\div 11 = 0.363636\ldots = 0.\overline{36}\).
Ex. 3 โ \(\tfrac{7}{8}\):
\(7\div 8 = 0.875\) (finite).
Ex. 4 โ \(\tfrac{12}{90}\):
\(12\div 90 = 0.1\overline{3}\) โ period โ3โ.
A fraction in irreducible form has a finite decimal expansion if and only if its denominator contains only factors 2 and/or 5.
Otherwise, the expansion is a repeating decimal.
1.10 ๐ ๐ง Solved Exercises โ Generating Fractions
1. \(0.\overline{3}\).
Let \(x=0.\overline{3}\). \(10x=3.\overline{3}\) โ \(9x=3\) โ \(x=\tfrac{1}{3}\).
2. \(0.\overline{72}\).
Let \(x=0.\overline{72}\). \(100x=72.\overline{72}\) โ \(99x=72\) โ \(x=\tfrac{72}{99}=\tfrac{8}{11}\).
3. \(2.\overline{1}\).
\(x=2.\overline{1}\). \(10x=21.\overline{1}\) โ \(9x=19\) โ \(x=\tfrac{19}{9}\).
4. \(0.4\overline{7}\).
\(x=0.4\overline{7}\). \(10x=4.\overline{7}\), \(100x=47.\overline{7}\) โ \(90x=43\) โ \(x=\tfrac{43}{90}\).
5. \(3.12\overline{5}\).
\(x=3.12555\ldots\). \(1000x=3125.555\ldots\), \(100x=312.555\ldots\) โ \(900x=2813\) โ \(x=\tfrac{2813}{900}\).
1.11 ๐งฉ Exercises โ Generating Fractions
- \(0.\overline{3}\), \(0.\overline{7}\), \(0.\overline{2}\)
- \(0.1\overline{3}\), \(0.72\overline{1}\), \(1.2\overline{45}\)
- \(0.\overline{81}\), \(2.\overline{6}\), \(3.4\overline{3}\)
- \(0.4\overline{5}\) (simplify)
1.12 โ Answer Key โ Generating Fractions
\(\tfrac{1}{3}\), \(\tfrac{7}{9}\), \(\tfrac{2}{9}\)
\(\tfrac{2}{15}\), \(\tfrac{649}{900}\), \(\tfrac{1233}{990}=\tfrac{137}{110}\)
\(\tfrac{9}{11}\), \(\tfrac{8}{3}\), \(\tfrac{103}{30}\)
\(\tfrac{41}{90}\)
1.13 ๐ Properties of Number Sets
๐ธ Useful definitions
Closure: a set is closed under an operation when the result remains in the set.
Ex.: \(\mathbb{N}\) is closed under addition; \(\mathbb{Z}\) is closed under subtraction.Density: a set is dense if between any two distinct elements there is another element of the same set.
Ex.: \(\mathbb{Q}\) is dense; also \(\mathbb{R}\setminus\mathbb{Q}\) is dense in \(\mathbb{R}\).
๐น Natural numbers \((\mathbb{N}=\{0,1,2,\ldots\})\)
Counting; closed under \(+\) and \(\cdot\); not closed under \(-\) or \(/\).
๐น Integers \((\mathbb{Z}=\{\ldots,-2,-1,0,1,2,\ldots\})\)
Include \(\mathbb{N}\); closed under \(+\), \(-\), \(\cdot\); not closed under \(/\).
๐น Rationals \((\mathbb{Q})\)
Fractions and finite/repeating decimals; closed under the four operations (with \(b\neq 0\)); dense.
๐น Irrationals \((\mathbb{R}\setminus\mathbb{Q})\)
Non-repeating infinite decimals; dense; not closed under \(+\) or \(\cdot\) (e.g., \(\sqrt{2}+(-\sqrt{2})=0\notin\) irrationals).
๐น Reals \((\mathbb{R})\)
\(\mathbb{Q}\cup(\mathbb{R}\setminus\mathbb{Q})\); represent the real line.
For \(a,b\in\mathbb{R}\), if \(b\neq 0\) then \(\tfrac{a}{b}\in\mathbb{R}\).
1.14 ๐ Summary โ Properties
| Set | Symbol | Main Properties |
|---|---|---|
| Natural numbers | \(\mathbb{N}\) | Closed under \(+\) and \(\cdot\); basis for counting. |
| Integers | \(\mathbb{Z}\) | Closed under \(+\), \(-\), \(\cdot\); include negatives. |
| Rationals | \(\mathbb{Q}\) | Fractions; finite or repeating decimals; dense. |
| Irrationals | \(\mathbb{R}\setminus\mathbb{Q}\) | Non-repeating decimals; dense. |
| Reals | \(\mathbb{R}\) | Union of rationals and irrationals; division defined for \(b\neq 0\). |
1.15 ๐ Basic Algebraic Properties in \(\mathbb{R}\)
- \(0\cdot x=0\);\((-x)\cdot y=-(x\cdot y)\);\((-x)(-y)=x\cdot y\)
- \(x+y=x\Rightarrow y=0\);\(x\cdot y=x\Rightarrow y=1\)
- \(x+y=0\Rightarrow y=-x\);\(x\cdot y=1\Rightarrow y=x^{-1}\,(x\neq 0)\)
- \(x+z=y+z\Rightarrow x=y\) (cancellation law for addition)
- \(z\neq 0\) and \(x\cdot z=y\cdot z\Rightarrow x=y\) (cancellation law for multiplication)
- \(x\neq 0,\ y\neq 0\Rightarrow x\cdot y\neq 0\) (there are no zero divisors in \(\mathbb{R}\))
1.16 โ๏ธ Auxiliary Operations
Subtraction: \(a-b=a+(-b)\).
Division (\(b\neq 0\)): \(\tfrac{a}{b}=a\cdot b^{-1}\).
Examples
- Additive inverse of \(5\): \(-5\) (since \(5+(-5)=0\)).
- Multiplicative inverse of \(3\): \(\tfrac{1}{3}\) (since \(3\cdot\tfrac{1}{3}=1\)).
- \(7-2=7+(-2)=5\); \(\ 5\div2=5\cdot\tfrac{1}{2}=\tfrac{5}{2}=2.5\).
1.17 ๐ง Review Exercises
- Classify: \(\sqrt{2}\), \(0.333\ldots\), \(\pi\), \(7/2\), \(\sqrt{25}\).
- Write the subsets of \(\mathbb{R}\) in order of inclusion.
- Give three examples of irrationals and justify.
- Differentiate between finite and infinite decimals with examples.
- Is \(3.727272\ldots\) rational? Find its generating fraction.
- Convert \(0.1666\ldots\) into a fraction.
- \(1.4142135\ldots\) (approx. \(\sqrt{2}\)) is it a repeating decimal? Why?
- If \(a+7=b+7\), what can you conclude about \(a\) and \(b\)?
- Simplify: \((-3)\cdot(-x)+(-2x)\).
- If \(x\cdot a=x\) and \(x\neq 0\), what can you conclude about \(a\)?
1.18 ๐ Solutions
- I, R, I, R, R.
- \(\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\).
- Ex.: \(\sqrt{3}\), \(\pi\), \(e\) (not fractions of integers; non-repeating decimal).
- \(1/4=0.25\) (finite) vs. \(1/3=0.\overline{3}\) (infinite repeating).
- \(x=3.7272\ldots\). \(100x=372.72\ldots\); \(100x-x=369\) โ \(x=\tfrac{369}{99}=\tfrac{41}{11}\).
- \(x=0.1666\ldots\). \(10x=1.666\ldots\) โ \(9x=1.5\) โ \(x=\tfrac{1.5}{9}=\tfrac{3}{18}=\tfrac{1}{6}\).
- No. It is infinite non-repeating โ irrational.
- Addition cancellation: \(a=b\).
- \((-3)(-x)=3x\) โ \(3x+(-2x)=x\).
- Multiplicative identity: \(a=1\).