The World of Numbers
⚑ Panic box
Number sets — $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$; irrationals complete $\mathbb{R}$ with $\mathbb{Q}$
Rational test — decimal terminates or repeats; irrational decimal is endless and non-repeating
Terminating test — lowest-form denominator has only prime factors $2$ and/or $5$
Between two rationals — use their average: $\dfrac{a+b}{2}$
1. Number sets & their origins
| Set | What belongs | Key relation / example |
|---|---|---|
| Natural $\mathbb{N}$ | Counting numbers | ${1,2,3,\ldots}$; excludes $0$ here |
| Integers $\mathbb{Z}$ | Negative numbers, zero, positives | ${\ldots,-2,-1,0,1,2,\ldots}$ |
| Rational $\mathbb{Q}$ | $\dfrac{p}{q}$, integers $p,q$, $q\ne0$ | Includes integers: $n=\dfrac{n}{1}$ |
| Irrational | Not expressible as $\dfrac{p}{q}$ | Examples: $\sqrt{2},\ \pi$ |
| Real $\mathbb{R}$ | All rational and irrational numbers | The complete number line |
History anchors — tallying used one-to-one matching · Lebombo bone: about 35,000 years old, 29 notches · Ishango bone: about 20,000 BCE; tallies include $11,13,17,19$ (primes from 10 to 20) and a doubling pattern · Vedic texts name powers of 10 up to $10^{12}$; Lalitavistara reaches $10^{53}$ · Brahmagupta formalised arithmetic with zero and negative numbers in 628 CE.
Zero & signed numbers — $0$ is the result of $a-a$ · positive = fortune ($\textit{dhana}$), negative = debt ($\textit{ṛiṇa}$) · $0$ is neither positive nor negative.
2. Integer & rational arithmetic
Zero laws — $a+0=a$ · $a-0=a$ · $a\times0=0$ · division by $0$ is undefined.
Integer sign rules
- Same signs in addition → add magnitudes; keep the sign.
- Different signs in addition → subtract magnitudes; keep the sign of the larger magnitude.
- Multiplication/division: same signs → positive; unlike signs → negative.
- Subtracting a negative → adding its positive: $a-(-b)=a+b$.
Rational-number rules
- Equivalent fractions — $\dfrac{a}{b}=\dfrac{c}{d}$ exactly when $ad=bc$; simplify by cancelling common factors.
- Add/subtract — use a common denominator: $\dfrac{a}{b}\pm\dfrac{c}{b}=\dfrac{a\pm c}{b}$.
- Multiply — $\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}$, with $b,d\ne0$.
- Divide — $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}$, with $b,d,c\ne0$.
- Properties — addition and multiplication are commutative; multiplication distributes: $p(q+r)=pq+pr$.
- Closure — rationals stay rational under $+,-,\times,\div$; division only when the divisor is non-zero.
3. Number line, distance & density
Locate $\dfrac{p}{q}$ — use an equivalent form with $q>0$ · divide each unit interval into $q$ equal parts · count $|p|$ parts from $0$, right for positive and left for negative. Improper fractions lie beyond $1$; locate them using a mixed number or equivalent intervals.
Absolute value — $|x|$ is distance from $0$; $|x|\ge0$ · distance between $a$ and $b$: $|a-b|$.
Density — for distinct rationals $a<b$, $\dfrac{a+b}{2}$ is rational and lies between them · repeat the averaging to find infinitely many rationals between them.
4. Irrationals & square-root lengths
Irrational number — cannot be written as a ratio of integers with non-zero denominator; its decimal neither ends nor repeats.
Proof pattern: $\sqrt{2}$ is irrational
- Assume $\sqrt{2}=\dfrac{p}{q}$ in lowest terms.
- Square and rearrange: $p^2=2q^2$; hence $p$ is even.
- Put $p=2k$: then $q^2=2k^2$; hence $q$ is even.
- Both even contradict lowest terms; reject the assumption.
Construct $\sqrt{2}$ — mark $OA=1$ on the number line · erect a perpendicular at $A$ and mark $AB=1$ · join $O$ to $B$; $OB=\sqrt{2}$ by Pythagoras · with centre $O$ and radius $OB$, mark the corresponding point on the line.
Pi — circumference ÷ diameter; irrational · $\pi\approx\dfrac{3927}{1250}=3.1416$ is an approximation, not an exact fraction · Madhava's series: $\pi=4\left(1-\dfrac13+\dfrac15-\dfrac17+\cdots\right)$.
5. Decimal patterns & conversions
| Number type | Decimal signature | How to identify / handle |
|---|---|---|
| Rational | Terminates or eventually repeats | In lowest form, denominator factors only $2,5$ → terminates; any other prime factor → repeats |
| Irrational | Endless, non-repeating | No repeating block; e.g. $\sqrt{2},\pi$ |
Terminating decimal places — for lowest-form denominator $2^m5^n$, decimal places needed = $\max(m,n)$; multiply numerator and denominator to make denominator a power of $10$.
Convert a repeating decimal to a fraction
- Set $x$ equal to the decimal.
- For a pure repeat of $n$ digits, form $10^n x$; subtract $x$.
- For a mixed repeat, first shift $m$ non-repeating digits with $10^m x$; then shift one full $n$-digit cycle with $10^n$ more.
- Subtract the shifted equations; solve for $x$ and simplify.
Cyclic pattern — $\dfrac17=0.\overline{142857}$; multiplying $142857$ by $1$ through $6$ rotates the same six digits.
Two decimal forms — a terminating decimal also has a form ending in repeating 9s: $1.000\ldots=0.999\ldots$.
6. Traps
$\mathbb{N}$ includes $0$→ in this chapter, $\mathbb{N}={1,2,3,\ldots}$.Every fraction is already simplest→ equivalent forms abound; reduce numerator and denominator by a common factor.A rational decimal must terminate→ it may repeat forever; check the lowest-form denominator.Any endless decimal is irrational→ it is rational if its digits eventually repeat.$\sqrt{2}$ is rational because it is a length→ its length cannot be written as a ratio of integers.$0.999\ldots<1$→ $0.999\ldots=1$.Divide by zero→ undefined; rational division also requires a non-zero divisor.