The Real Number Line
In your high school lab, a temperature sensor reads $25.0^\circ\text{C}$, but it never gives "the" exact value: it always has an uncertainty of $0.2^\circ\text{C}$. How do you write, without ambiguity, all the values the real temperature could be? With intervals, neighborhoods, and absolute value, the language you will use in physics, chemistry, and in the calculus of the coming semesters.
Intervals and inequalities
Every real number is a point on the line, and an inequality marks a stretch of that line. An interval is the shorthand way to write that stretch.
For example, $x \ge -3$ and $x < 5$ is written as the double inequality $-3 \le x < 5$ and as the interval $[-3, 5)$. On the number line, $-3$ gets a filled dot and $5$ an open dot.
Express as an interval and as an inequality the set of real numbers that are greater than or equal to $-3$ and strictly less than $5$.
Express as an interval and as an inequality the set of real numbers that are greater than or equal to $-3$ and strictly less than $5$.
Write as an interval the set of real numbers satisfying the condition $|x| \le 7$.
Write as an interval the set of real numbers satisfying the condition $|x| \le 7$.
State which of the following real numbers belong to the interval $[-1, \pi)$: $-1.01$, $-1$, $3.14$, $\frac{22}{7}$, $\sqrt{10}$.
State which of the following real numbers belong to the interval $[-1, \pi)$:
$-1.01$, $-1$, $3.14$, $\frac{22}{7}$, $\sqrt{10}$.
Neighborhoods: E(a, r)
When a piece of data is given as "a central value with a tolerance," the interval is centered at that value. This form is called a neighborhood, and you will see it again when you study limits.
To go from an interval to a neighborhood: the center is the midpoint, $a = \frac{\text{left} + \text{right}}{2}$, and the radius is half the length, $r = \frac{\text{right} - \text{left}}{2}$.
Express the neighbourhood $E(-1, 3)$ as a numerical interval and describe it using an absolute value inequality.
Express the neighbourhood $E(-1, 3)$ as a numerical interval and describe it using an absolute value inequality.
Given the open interval $(2, 8)$, express it in neighborhood notation, explicitly indicating its center $a$ and its radius $r$.
Given the open interval $(2, 8)$, express it in neighborhood notation, explicitly indicating its center $a$ and its radius $r$.
The price of a stock on the stock market fluctuates around a central value. If the fluctuation interval during the week was $[112.5 , 127.5]$, express this range using…
The price of a stock on the stock market fluctuates around a central value. If the fluctuation interval during the week was $[112.5 , 127.5]$, express this range using neighbourhood notation $E(a, r)$ and using an absolute value inequality of the form $|x - a| \le r$.
Distance and equations with absolute value
With this idea, an equation like $|x - 4| = 9$ reads: "Which numbers are at distance $9$ from the number $4$?"
Calculate the distance on the number line between points $A = -4.5$ and $B = 2.3$, and express the mathematical operation using absolute value notation.
Calculate the distance on the number line between points $A = -4.5$ and $B = 2.3$, and express the mathematical operation using absolute value notation.
Find the values of $x$ that satisfy the equation $|x - 4| = 9$.
Find the values of $x$ that satisfy the equation $|x - 4| = 9$.
Tolerances: absolute value in inequalities
The two ways of reading an absolute value with an inequality:
In a biology laboratory, an incubator must maintain a temperature $T$ in degrees Celsius that does not differ from $37^{\circ}\text{C}$ by more than…
In a biology laboratory, an incubator must maintain a temperature $T$ in degrees Celsius that does not differ from $37^{\circ}\text{C}$ by more than $0.5^{\circ}\text{C}$. Express this tolerance condition using an absolute value inequality and determine the interval of permitted temperatures.
A cylindrical engine part has an ideal diameter of $50\text{ mm}$. Quality control rejects the part if the measured diameter $d$ satisfies $|d - 50| > 0.02$. Write the…
A cylindrical engine part has an ideal diameter of $50\text{ mm}$. Quality control rejects the part if the measured diameter $d$ satisfies $|d - 50| > 0.02$. Write the interval of diameters that are accepted by quality control.
In an archery competition, a shot is considered "excellent" if the arrow hits at a distance $d$ from the center of the target that satisfies $|d| < 2.5\text{ cm}$. If…
In an archery competition, a shot is considered "excellent" if the arrow hits at a distance $d$ from the center of the target that satisfies $|d| < 2.5\text{ cm}$. If the target is positioned such that its center is the origin of coordinates on a one-dimensional axis, state the interval of "excellent" positions.
Union, intersection, and complement
Since intervals are sets, they can be combined:
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Union $A \cup B$: the numbers that are in $A$, in $B$, or in both.
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Intersection $A \cap B$: the numbers that are in $A$ and in $B$.
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Complement $\mathbb{R} \setminus I$: the reals that are not in $I$.
The safest way to operate is to draw each interval on a number line, one below the other, and see what remains.
Given the intervals $A = [-2, 4)$ and $B = (1, 6]$, determine and express as a single interval the result of the union $A \cup B$ and of the intersection $A \cap B$.
Given the intervals $A = [-2, 4)$ and $B = (1, 6]$, determine and express as a single interval the result of the union $A \cup B$ and of the intersection $A \cap B$.
Given the sets $A = (-\infty, 2]$, $B = (-1, 5)$ and $C = [0, +\infty)$, calculate $(A \cap B) \cup C$ and express the result as a single numerical interval.
Given the sets $A = (-\infty, 2]$, $B = (-1, 5)$ and $C = [0, +\infty)$, calculate $(A \cap B) \cup C$ and express the result as a single numerical interval.
Consider the interval $I = [-3, 4)$. Determine the complementary set $\mathbb{R} \setminus I$ and express it as the union of two intervals.
Consider the interval $I = [-3, 4)$. Determine the complementary set $\mathbb{R} \setminus I$ and express it as the union of two intervals.
Absolute value inequalities
They are solved with the property from the tolerances section: you remove the absolute value with a double inequality and solve for $x$ across the whole chain.
When there are two absolute values, it helps to think in terms of distances.
For nested absolute values, you solve from the outside in.
Solve the inequality $|2x - 5| \le 3$ and represent the solution set as an interval.
Solve the inequality $|2x - 5| \le 3$ and represent the solution set as an interval.
Find the set of real numbers $x$ that satisfy the inequality $|x + 2| + |x - 3| \le 7$. Express the result as an interval. The figure shows the graph of the function…
Find the set of real numbers $x$ that satisfy the inequality $|x + 2| + |x - 3| \le 7$. Express the result as an interval.
The figure shows the graph of the function $f(x) = |x + 2| + |x - 3|$ and the line $y = 7$, so that you can check your result.
Solve the inequality with nested absolute values $||x - 1| - 2| \le 1$ and express the final solution as the union of two closed intervals.
Solve the inequality with nested absolute values $||x - 1| - 2| \le 1$ and express the final solution as the union of two closed intervals.
Parameters and bounds
There are problems where an endpoint depends on a parameter, or where a measurement with error is translated into another quantity. It is always done the same way: translate into inequalities, draw on the number line, and interpret.
Determine the values of the parameter $k \in \mathbb{R}$ for which the intersection of the intervals $A = (-\infty, 2k]$ and $B = (k + 3, +\infty)$ is an interval of…
Determine the values of the parameter $k \in \mathbb{R}$ for which the intersection of the intervals $A = (-\infty, 2k]$ and $B = (k + 3, +\infty)$ is an interval of length exactly equal to $5$.
In an optics experiment, the position $x$ of a lens along a one-dimensional rail must simultaneously satisfy two conditions for the image to be in focus: the distance to…
In an optics experiment, the position $x$ of a lens along a one-dimensional rail must simultaneously satisfy two conditions for the image to be in focus: the distance to the light source $A$ (located at $x=10$) must be greater than $5\text{ cm}$, and the distance to the screen $B$ (located at $x=30$) must be less than $12\text{ cm}$. Express these conditions using a system of absolute value inequalities and determine the valid interval for the position $x$.
The measurement of the radius of a sphere is $r = 5.0 \pm 0.1\text{ cm}$. Using the properties of inequalities and intervals on the number line, determine the strict…
The measurement of the radius of a sphere is $r = 5.0 \pm 0.1\text{ cm}$. Using the properties of inequalities and intervals on the number line, determine the strict interval in which the exact volume of the sphere, $V = \frac{4}{3}\pi r^3$, lies. Consider the value of $\pi$ unapproximated at the endpoints of the resulting interval.
Irrational numbers on the number line
Irrational numbers also take up a place on the line, and sometimes they can be located exactly with a straightedge and compass using the Pythagorean theorem. Also, between any two rational numbers there are always irrational numbers.
Locating $\sqrt{13}$
If you are looking for a number that is the root of a sum of squares, use a right triangle. Since $13 = 9 + 4 = 3^2 + 2^2$, the hypotenuse of a triangle with legs $3$ and $2$ measures $\sqrt{13}$.
The golden ratio
A similar construction gives $\phi = \frac{1 + \sqrt{5}}{2}$. Start with a square of side $1$. Mark the midpoint $M$ of its base, join $M$ to the upper right vertex, and, with center at $M$, draw an arc to the extension of the base.
An irrational between two rationals
Explain the steps to represent the irrational number $\sqrt{13}$ exactly on the number line using Pythagoras' theorem.
Explain the steps to represent the irrational number $\sqrt{13}$ exactly on the number line using Pythagoras' theorem.
In the design of a building facade, the golden ratio $\phi = \frac{1 + \sqrt{5}}{2}$ is used. Based on the attached figure, analytically justify why the construction…
In the design of a building facade, the golden ratio $\phi = \frac{1 + \sqrt{5}}{2}$ is used. Based on the attached figure, analytically justify why the construction shown determines exactly the value of $\phi$ on the number line.
Prove, by finding an explicit example, that between the rational numbers $\frac{10}{7}$ and $\frac{143}{100}$ there is at least one irrational number.
Prove, by finding an explicit example, that between the rational numbers $\frac{10}{7}$ and $\frac{143}{100}$ there is at least one irrational number.
Back to the beginning: the sensor that reads $25.0^\circ\text{C}$ with an uncertainty of $0.2^\circ\text{C}$ describes all its possible values with the neighborhood $E(25, 0.2)$, with the inequality $|T - 25| < 0.2$, or with the interval $(24.8, 25.2)$.