Type $1 \div 3$ into a calculator and you see $0.3333333333$. Type $\sqrt{2}$ and you see $1.414213562$. Both look like ordinary decimals, but one of them is exactly a fraction and the other never can be. Which is which, and how can you tell? By the end of this page you will know, and you will be able to describe any set of real numbers precisely with interval notation.
The number sets and how they nest
Numbers come in families, and each family sits inside a bigger one.
Some US books start $\mathbb{N}$ at $1$ and call $\{0, 1, 2, \dots\}$ the whole numbers. We include $0$ in $\mathbb{N}$ here. The classification of every example below works either way, except for $0$ itself.
The inclusions are strict: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$. Also $\mathbb{Q} \cap \mathbb{I} = \emptyset$ and $\mathbb{Q} \cup \mathbb{I} = \mathbb{R}$. A number can belong to several sets at once, so we classify it in the most restrictive (smallest) set that contains it.
To classify a number, work down this list:
Simplify it first.
Is it a whole number, $0, 1, 2, \dots$? Then it is in $\mathbb{N}$.
Is it a whole number or its opposite, like $-4$? Then it is in $\mathbb{Z}$.
Is it a ratio of integers, or a decimal that terminates or repeats? Then it is in $\mathbb{Q}$.
Otherwise it is irrational.
Numbers in scientific notation are no exception. $2.5 \times 10^{-3} = 0.0025 = \dfrac{1}{400}$ terminates, so it is rational.
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Classify the following numbers by indicating which of the number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$ they belong to (bear in mind that a number…
Classify the following numbers by indicating which of the number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$ they belong to (bear in mind that a number can belong to several sets at the same time):
In a Physics class, different quantities are measured or used for an experiment: the mass of a particle $m = 2.5 \cdot 10^{-3}$ pounds, the period of a pendulum…
In a Physics class, different quantities are measured or used for an experiment: the mass of a particle $m = 2.5 \cdot 10^{-3}$ pounds, the period of a pendulum $T = 2\pi$ s, the number of oscillations $n = 15$, the change in temperature $\Delta \theta = -4^\circ$°F and the energy $E = \sqrt{2}$ J. Classify the exact numerical values of these measurements according to the most restrictive (smallest) number set they belong to ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$).
Every rational number is a decimal that either terminates ($0.75$) or repeats ($0.\overline{27}$). The reverse is also true: every terminating or repeating decimal is a fraction.
Terminating decimals are easy. Read the place value: $3.84 = \dfrac{384}{100} = \dfrac{96}{25}$ in simplest form.
Repeating decimals need a trick. Some problems call the fraction you get the generating fraction.
Once you have fractions, arithmetic is easy. To add repeating decimals, convert each, add with a common denominator, and convert back if asked. For example, $1.\overline{2} + 0.\overline{5} = \dfrac{11}{9} + \dfrac{5}{9} = \dfrac{16}{9} = 1.\overline{7}$.
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Find the irreducible generating fraction of the following decimal numbers: a) $2.45$ b) $1.\widehat{6}$ c) $0.1\widehat{25}$
Find the irreducible generating fraction of the following decimal numbers:
State whether the following statements are true or false, briefly justifying your answer: a) Every integer is a rational number. b) The number $0$ is an irrational…
State whether the following statements are true or false, briefly justifying your answer:
a) Every integer is a rational number.
b) The number $0$ is an irrational number.
c) All decimal numbers can be expressed as a fraction.
An architect calculates the length of a wall by adding two sections whose theoretical measurements are $L_1 = 2.\widehat{3}$ ft and $L_2 = 1.1\widehat{6}$ ft. a)…
An architect calculates the length of a wall by adding two sections whose theoretical measurements are $L_1 = 2.\widehat{3}$ ft and $L_2 = 1.1\widehat{6}$ ft.
a) Calculate the exact total length of the wall by operating with the generating fractions of both numbers.
How do we know $\sqrt{2}$ or $\sqrt{3}$ is irrational? Looking at digits on a calculator can never prove it, since we only see finitely many. We need an argument. The standard tool is proof by contradiction: assume the opposite, and show it leads to something impossible.
Operations with rational and irrational numbers
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Without using a calculator, determine whether the result of the following operations is a rational or irrational number by expanding the algebraic expressions: a)…
Without using a calculator, determine whether the result of the following operations is a rational or irrational number by expanding the algebraic expressions:
Prove, using proof by contradiction, that $\sqrt{3}$ is an irrational number. (Assume that $\sqrt{3} = \frac{p}{q}$ where $p$ and $q$ are coprime integers, and reach a…
Prove, using proof by contradiction, that $\sqrt{3}$ is an irrational number. (Assume that $\sqrt{3} = \frac{p}{q}$ where $p$ and $q$ are coprime integers, and reach a contradiction).
Irrational numbers are best handled exactly, with radicals, not with decimals. Two techniques come up often: verifying that a number satisfies an equation, and simplifying nested radicals.
Numbers that satisfy an equation
The number $s$ above is a zero of $y = x^2 - 2x - 1$. Look for it where the graph crosses the $x$-axis:
The graph crosses at $1 - \sqrt{2} \approx -0.41$ and $1 + \sqrt{2} \approx 2.41$.
Nested radicals
An expression such as $\sqrt{a - b\sqrt{c}}$ can often be simplified by recognizing the radicand as a perfect square of a binomial: $(m - n\sqrt{c})^2 = m^2 + n^2c - 2mn\sqrt{c}$.
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The golden ratio is defined as $\phi = \frac{1+\sqrt{5}}{2}$. a) Prove algebraically that $\phi$ satisfies the equation $\phi^2 - \phi - 1 = 0$. b) From this result,…
The golden ratio is defined as $\phi = \frac{1+\sqrt{5}}{2}$.
a) Prove algebraically that $\phi$ satisfies the equation $\phi^2 - \phi - 1 = 0$.
b) From this result, deduce whether the number $\frac{1}{\phi}$ is rational or irrational and express it as a binomial in terms of $\phi$.
Consider the real number $x = \sqrt{7 - 4\sqrt{3}} + \sqrt{3}$. Simplify the expression fully (hint: try to express the radicand $7 - 4\sqrt{3}$ as a perfect square…
Consider the real number $x = \sqrt{7 - 4\sqrt{3}} + \sqrt{3}$.
Simplify the expression fully (hint: try to express the radicand $7 - 4\sqrt{3}$ as a perfect square binomial) and determine which number sets ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$) the exact value of $x$ belongs to.
Can you actually mark $\sqrt{13}$ on the number line? Yes, with a right triangle, the Pythagorean Theorem and a compass.
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The figure shows the geometric construction of a number on the real line using a right-angled triangle and an arc of a circle. a) Identify which exact real number the…
The figure shows the geometric construction of a number on the real line using a right-angled triangle and an arc of a circle.
a) Identify which exact real number the point $P$ represents on the horizontal axis.
b) Classify this number in the number sets.
c) Prove its value analytically by applying Pythagoras' theorem.
Order the following real numbers from smallest to largest. To do this, you can use their approximate decimal expression: $3.14$, $\pi$, $\sqrt{10}$, $3.1\widehat{4}$,…
Order the following real numbers from smallest to largest. To do this, you can use their approximate decimal expression:
A company divides a profit of $\$1000$ equally among $3$ partners. The accounting software truncates the result and transfers $\$333.33$ to each account. a) Express the…
A company divides a profit of $\$1000$ equally among $3$ partners. The accounting software truncates the result and transfers $\$333.33$ to each account.
a) Express the exact amount corresponding to each partner using fractions.
b) Calculate the total absolute error made by the company when using the decimal approximation for the three partners instead of the exact value.
Pick two real numbers as close together as you like. There is always room for more.
Two practical tools for finding numbers between $a < b$:
Rational: write both fractions with a common denominator, then multiply numerator and denominator by a bigger number to create gaps between the numerators.
Irrational: take $a + (b-a)\cdot\dfrac{\sqrt{2}}{2}$. Since $0 < \dfrac{\sqrt{2}}{2} < 1$, this lies between $a$ and $b$, and it is irrational when $a, b$ are rational. Or build a decimal whose digits never repeat, like $0.3010010001\dots$ (one more $0$ each time).
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Find a rational number and an irrational number that are strictly between $1.5$ and $1.6$.
Find a rational number and an irrational number that are strictly between $1.5$ and $1.6$.
The density property of rational numbers states that between any two distinct rational numbers there is always another rational number. Prove this property by finding…
The density property of rational numbers states that between any two distinct rational numbers there is always another rational number. Prove this property by finding three distinct rational numbers strictly between $\frac{4}{7}$ and $\frac{5}{7}$.
Knowing that between any two real numbers there are infinitely many rational and irrational numbers, construct an irrational number $x$ that simultaneously meets the…
Knowing that between any two real numbers there are infinitely many rational and irrational numbers, construct an irrational number $x$ that simultaneously meets the following two conditions:
a) $0.01 < x < 0.02$
b) In its infinite decimal expansion, only the digits $0$ and $1$ appear.
Write at least the first 12 decimal places of your number to make clear the formation rule that guarantees its irrationality.
Writing "all real numbers from $-1$ up to, but not including, $3$" over and over is slow. Interval notation is the shorthand.
The same set can be written three ways: $\{x \in \mathbb{R} \mid -1 \le x < 3\}$, the interval $[-1, 3)$, and the graph above.
To list the integers in an interval, write them with braces (set notation). For $[-1, 3)$ the integers are $\{-1, 0, 1, 2\}$. The endpoint $3$ is excluded, and decimals like $0.5$ are not integers.
Union $A \cup B$ is everything in $A$ or $B$. Intersection $A \cap B$ is everything in both. Draw both on one number line and read off the result.
If two intervals do not overlap, the union is written with the symbol $\cup$ between them, for example $(-\infty, -1) \cup [2, \infty)$.
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Write as an interval and represent graphically on the real line the following set of numbers: $A = \{x \in \mathbb{R} \mid -2 < x \le 5\}$
Write as an interval and represent graphically on the real line the following set of numbers:
Engineers and manufacturers specify a target value plus or minus a tolerance. The absolute value describes exactly that.
Quadratic inequalities
To solve $ax^2 + bx + c \le 0$ (or $<$, $\ge$, $>$): factor, find the zeros, and use the parabola. If it opens up, it is below the $x$-axis between the zeros and above outside them.
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In the design of a microchip, the allowable tolerance for the thickness of a silicon track is given by the set $T = \{x \in \mathbb{R} \mid |x - 1.25| \le 0.05\}$ in…
In the design of a microchip, the allowable tolerance for the thickness of a silicon track is given by the set $T = \{x \in \mathbb{R} \mid |x - 1.25| \le 0.05\}$ in micrometers.
a) Express this set as an interval.
b) Determine whether the thicknesses $x_1 = 1.30$ and $x_2 = 1.25 + \frac{\sqrt{2}}{100}$ meet the required tolerance.
Given the sets of real numbers $A = \{x \in \mathbb{R} \mid x^2 - 5x + 6 \le 0\}$ and $B = \{x \in \mathbb{R} \mid |x - 2| < 1\}$, find the resulting sets of the…
Given the sets of real numbers $A = \{x \in \mathbb{R} \mid x^2 - 5x + 6 \le 0\}$ and $B = \{x \in \mathbb{R} \mid |x - 2| < 1\}$, find the resulting sets of the operations $A \cup B$ and $A \cap B$. Express the final results as an interval.
When a quantity must satisfy several conditions at the same time, the answer is an intersection. Taking the largest left endpoint and the smallest right endpoint gives the result. If an endpoint is shared, the open one wins.
When does an intersection have exactly one point?
To place an irrational number inside an interval, use a root whose radicand lies between the squares of the endpoints. In $[4, 6]$, take $\sqrt{17}$, since $4^2 = 16 < 17 < 36 = 6^2$ and $17$ is not a perfect square. For small decimal intervals, scale a root: $\dfrac{\sqrt{n}}{10}$ or $\dfrac{\sqrt{n}}{100}$.
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A chemical reaction is viable if the temperature $T$ (in $^\circ$°F) belongs to the interval $A = [-10, 45)$ and, simultaneously, the pressure forces a temperature…
A chemical reaction is viable if the temperature $T$ (in $^\circ$°F) belongs to the interval $A = [-10, 45)$ and, simultaneously, the pressure forces a temperature control parameter $k = \frac{T}{10}$ to belong to the interval $B = (0, 5]$. Calculate the exact interval of temperatures $T$ for which the reaction is viable, satisfying both conditions at the same time.
Given the sets defined on the real line $A = (-\infty, 3k-1]$ and $B = [k+5, \infty)$, determine the value of the parameter $k \in \mathbb{R}$ for which the intersection…
Given the sets defined on the real line $A = (-\infty, 3k-1]$ and $B = [k+5, \infty)$, determine the value of the parameter $k \in \mathbb{R}$ for which the intersection $A \cap B$ consists of a single real number. Classify this number in the sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$.
A sports analyst models an athlete's optimal performance through the intersection of three reaction time intervals (in seconds) obtained from different biomechanical…
A sports analyst models an athlete's optimal performance through the intersection of three reaction time intervals (in seconds) obtained from different biomechanical studies:
$I_1 = [0.15, 0.25]$
$I_2 = (0.18, 0.30)$
$I_3 = [0.20, \infty)$
a) Calculate the resulting interval of the intersection $I_1 \cap I_2 \cap I_3$.
b) If the absolute ideal reaction time must be an irrational number strictly contained within that intersection, propose an exact value (using roots) that meets the condition.
Back to the start: $\dfrac{1}{3} = 0.\overline{3}$ is exactly a fraction, so it is rational. Its calculator display $0.3333333333$ is a rounded copy. $\sqrt{2}$ is irrational (by contradiction), so no decimal and no fraction ever equals it. The calculator's $1.414213562$ is only an approximation.