Irrational Numbers
Emma types $\sqrt{2}$ into her calculator and gets $1.41421356\ldots$ Then she squares $1.4142$ to check it and gets $1.99996164$, close to $2$ but not $2$. She tries more digits and still never hits $2$ exactly. Is the calculator broken? No. $\sqrt{2}$ really can never be written as a fraction or as a decimal that ends or repeats. By the end of this page you will see why.
What is an irrational number?
Every rational number can be written as a fraction $\frac{a}{b}$ of integers ($b \neq 0$). Its decimal either terminates ($0.75$) or repeats ($0.\overline{3}$). The reason is long division: each remainder is smaller than $b$, so after at most $b$ steps a remainder repeats and the digits start cycling.
Some numbers break that pattern.
You will meet three kinds of irrational numbers in this course:
-
Roots of non-perfect squares, such as $\sqrt{2}$, $\sqrt{7}$, $\sqrt{13}$.
-
Famous constants, such as $\pi \approx 3.14159\ldots$ and Euler's number $e \approx 2.71828\ldots$
-
Decimals built with no repeating pattern, such as $0.1010010001\ldots$ (the run of zeros keeps growing).
Classify the following numbers by indicating which belong to the set of irrational numbers and which do not. Justify your answer based on their decimal expression: a)…
Classify the following numbers by indicating which belong to the set of irrational numbers and which do not. Justify your answer based on their decimal expression:
- a) $3.14159$
- b) $\sqrt{7}$
- c) $\frac{22}{7}$
- d) $1.1010010001...$
- e) $\sqrt{25}$
Identify the pattern followed by the decimal digits of the following number and explain why it is considered an irrational number: $x = 0.12345678910111213...$
Identify the pattern followed by the decimal digits of the following number and explain why it is considered an irrational number:
$x = 0.12345678910111213...$
Consider the sequence of numbers formed by writing zeros and ones in the following way: we start with a $0.1$, then we add a $0$ and a $1$, then two $0$s and a $1$, then…
Consider the sequence of numbers formed by writing zeros and ones in the following way: we start with a $0.1$, then we add a $0$ and a $1$, then two $0$s and a $1$, then three $0$s and a $1$, and so on to infinity.
The resulting number is $N = 0.101001000100001...$
Analyze the structure of this number and justify reasonably why $N$ belongs to the set of irrational numbers and cannot be expressed as a fraction $\frac{a}{b}$.
Why $\sqrt{2}$ is irrational
How can anyone be sure $\sqrt{2}$ will never turn into a fraction, no matter how many digits we check? We prove it. The method is proof by contradiction: assume the opposite, then show that this leads to something impossible.
The same idea works for any whole number.
Use the slider to see which square roots come out whole.
Determine whether the following statements are true or false. a) All decimal numbers with infinite digits are irrational. b) The square root of any natural number is an…
Determine whether the following statements are true or false.
- a) All decimal numbers with infinite digits are irrational.
- b) The square root of any natural number is an irrational number.
- c) The number $\pi$ can be written exactly as a fraction.
- d) The sum of a rational number and an irrational number is always irrational.
Locating irrational numbers with the Pythagorean Theorem
$\sqrt{2}$ is not a fraction, but it is a real point on the number line. You can build it exactly: draw a right triangle with legs $1$ and $1$. By the Pythagorean Theorem its hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$. Then swing a compass arc from the hypotenuse down to the number line.
If a leg is itself a square root, it works the same way, because $(\sqrt{a})^2 = a$. Start with legs $1$ and $1$ to get $\sqrt{2}$. Use that hypotenuse and a new leg of $1$ to get $\sqrt{2 + 1} = \sqrt{3}$. Keep going and you get a whole spiral of square roots.
To represent the irrational number $\sqrt{13}$ on the number line using the Pythagorean theorem, we need to find two integers whose squares add up to $13$. State what…
To represent the irrational number $\sqrt{13}$ on the number line using the Pythagorean theorem, we need to find two integers whose squares add up to $13$. State what these two numbers are and draw a mental sketch of how you would form the right triangle.
Observe the following graphical representation on the number line. A right triangle has been constructed resting on the horizontal axis. Calculate the exact value of the…
Observe the following graphical representation on the number line. A right triangle has been constructed resting on the horizontal axis.
Calculate the exact value of the hypotenuse $h$. What irrational number does the red dot $X$ represent on the number line?
The famous "Spiral of Theodorus" is constructed by drawing contiguous right triangles. The first has legs of length $1$. The hypotenuse of this triangle serves as a leg…
The famous "Spiral of Theodorus" is constructed by drawing contiguous right triangles. The first has legs of length $1$. The hypotenuse of this triangle serves as a leg for the next one, whose other leg again measures $1$, and so on.
Calculate the exact value of the hypotenuses $h_1$, $h_2$ and $h_3$. Following this logical pattern, what will be the exact value of the hypotenuse $h_8$ of the eighth triangle? Is it rational or irrational?
Diagonals and distances
Many real lengths turn out irrational because they are hypotenuses. A diagonal of a rectangle with sides $a$ and $b$ is $\sqrt{a^2 + b^2}$. Give the exact length as a radical, then approximate it when you need something you can measure.
For a box, do it in two steps. Find the diagonal of the base first, then use it as one leg of a second right triangle with the height as the other leg.
A soccer field has dimensions of $100$ ft long by $50$ ft wide. The maintenance team needs to lay an irrigation pipe that crosses the field diagonally. Calculate the…
A soccer field has dimensions of $100$ ft long by $50$ ft wide. The maintenance team needs to lay an irrigation pipe that crosses the field diagonally. Calculate the exact length of the pipe using radicals. Is this length a rational or an irrational number? Approximate the result to two decimal places to find out how many feet of pipe they must buy.
In baseball, the bases form a perfect square with sides of $90$ feet ($27.43$ feet). When the catcher throws the ball from home plate to second base to prevent a steal,…
In baseball, the bases form a perfect square with sides of $90$ feet ($27.43$ feet). When the catcher throws the ball from home plate to second base to prevent a steal, the ball crosses the field diagonally. Calculate the exact distance in feet that the ball travels. Is it a rational or an irrational quantity?
Observe the following cube of side $a = 5$ in. The diagonal of one of its faces is called $d$, and the main diagonal that crosses the inside of the cube is called $D$.…
Observe the following cube of side $a = 5$ in. The diagonal of one of its faces is called $d$, and the main diagonal that crosses the inside of the cube is called $D$.
Calculate the exact value of the diagonal of the base ($d$) and, using it as a leg along with the height of the cube, calculate the exact length of the main diagonal ($D$). Is it possible for a cube with an integer edge to have a main diagonal that is a rational number?
Exact values vs. approximations
$\sqrt{19}$ and $\pi$ have endless decimals, so on a ruler or a calculator screen you can only use an approximation. The exact value is the symbol itself: $\sqrt{19}$, $\pi$. Write $=$ for exact values and $\approx$ for approximations.
There are two ways to cut a decimal short:
-
Truncate: chop off the digits after a place and do not change what is left.
-
Round: look at the next digit. If it is $5$ or more, round the last kept digit up. Otherwise leave it.
To see how good an approximation is, find the absolute error: $|\text{approximate} - \text{exact}|$.
Truncate and round the irrational number $\sqrt{11} = 3.31662479...$ to two and three decimal places. Organize the results in a table.
Truncate and round the irrational number $\sqrt{11} = 3.31662479...$ to two and three decimal places. Organize the results in a table.
The golden ratio, represented by the Greek letter $\Phi$ (Phi), has an exact value of $\frac{1+\sqrt{5}}{2}$. Use your calculator to find its decimal expression and…
The golden ratio, represented by the Greek letter $\Phi$ (Phi), has an exact value of $\frac{1+\sqrt{5}}{2}$. Use your calculator to find its decimal expression and write down the first six decimal digits. Is it a rational or an irrational number?
In a mechanical parts factory, a circular plate with a radius of exactly $40$ in must be cut. The automatic lathe requires the circumference of the circle to be entered…
In a mechanical parts factory, a circular plate with a radius of exactly $40$ in must be cut. The automatic lathe requires the circumference of the circle to be entered to make the cut.
- a) What is the exact circumference of the piece?
- b) If the operator enters the approximate value using $\pi = 3.14$, what is the absolute error made compared to the real value (using all the digits of $\pi$ on the calculator)?
Operations with irrational numbers
Does adding or multiplying irrational numbers give an irrational number? It depends. First, here are two rules about mixing rational and irrational numbers, and two tools for products of radicals.
Two irrational numbers can combine either way, so you must check each case.
Evaluate the result of the following operations and classify the final result as rational or irrational: a) $\sqrt{2} \cdot \sqrt{8}$ b) $\sqrt{3} + \sqrt{3}$ c)…
Evaluate the result of the following operations and classify the final result as rational or irrational:
- a) $\sqrt{2} \cdot \sqrt{8}$
- b) $\sqrt{3} + \sqrt{3}$
- c) $(\sqrt{5})^2$
- d) $\pi - \pi$
A technology company is designing a new television screen. They want the ratio of width to height to be exactly $\sqrt{2}$ so that, when dividing the screen in half, the…
A technology company is designing a new television screen. They want the ratio of width to height to be exactly $\sqrt{2}$ so that, when dividing the screen in half, the two resulting rectangles maintain the same proportion (as happens with letter-sized paper). If the height of the screen is going to be $50$ in, what must its exact width be? Approximate the result to the nearest tenth of an inch.
In a math class, the teacher proposes the following challenge: "If we add an irrational number to another irrational number, is the result always irrational?". Find two…
In a math class, the teacher proposes the following challenge: "If we add an irrational number to another irrational number, is the result always irrational?".
Find two different irrational numbers whose sum is an integer rational number and show with simple algebraic operations that this is possible.
Exact answers in formulas
Many formulas contain $\pi$ or a root. Substitute the numbers, simplify as far as possible, and leave the answer exact. Then approximate only if the question asks for it.
An answer such as $64\pi$ in.$^2$ is exact. The area of a $16$ in. pizza (radius $8$ in.) is $\pi \cdot 8^2 = 64\pi \approx 201.06$ in.$^2$.
The golden ratio is often used in art and architecture. A picture frame has been designed to be a "golden rectangle". If the shorter side of the frame measures $20$ in,…
The golden ratio is often used in art and architecture. A picture frame has been designed to be a "golden rectangle". If the shorter side of the frame measures $20$ in, the longer side must measure $20 \cdot \Phi$ in, where $\Phi = \frac{1+\sqrt{5}}{2}$. Calculate the exact length of the longer side and its approximation in inches rounded to one decimal place.
The period $T$ (in seconds) of a simple pendulum, that is, the time it takes to make one complete oscillation, is calculated using the formula…
The period $T$ (in seconds) of a simple pendulum, that is, the time it takes to make one complete oscillation, is calculated using the formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the string in feet and $g$ is the acceleration due to gravity ($9.8$ ft/s$^2$). Calculate the exact period of a pendulum of length $9.8$ feet. Is the result a rational or an irrational number?
To build an artificial honeycomb, a beekeeper designs cells in the shape of a regular hexagon. Knowing that the area of a regular hexagon of side $L$ is calculated using…
To build an artificial honeycomb, a beekeeper designs cells in the shape of a regular hexagon. Knowing that the area of a regular hexagon of side $L$ is calculated using the formula $A = \frac{3\sqrt{3}}{2} L^2$, calculate the exact area of a cell if its side measures $4$ in. Is it a rational or an irrational number?
Ordering and comparing real numbers
Between any two real numbers there are infinitely many rational and infinitely many irrational numbers. So the number line has no gaps: every point is a real number, and every real number is a point.
To compare numbers of different types, write each as a decimal (to enough places) and line them up. The next number line zooms in on $3.14$ through $3.17$.
To compare a root with a decimal without a calculator, square both numbers (they are positive). Is $\sqrt{10}$ greater than $3.2$? We have $3.2^2 = 10.24 > 10$, so $3.2 > \sqrt{10}$.
Small differences in a constant can grow. For example, $2.7^2 = 7.29$ while $e^2 \approx 7.389$. The gap gets bigger as the exponent gets bigger, so in a formula with $e^t$ use the calculator's $e$, not a rounded value.
Order the following real numbers from smallest to largest. To do this, you can use their decimal approximations: $3.14$ , $\pi$ , $\sqrt{10}$ , $\frac{22}{7}$ , $3.1416$
Order the following real numbers from smallest to largest. To do this, you can use their decimal approximations:
$3.14$ , $\pi$ , $\sqrt{10}$ , $\frac{22}{7}$ , $3.1416$
An investor analyzes the growth of a bacteria culture used to manufacture bioplastics. The population in millions follows the formula $P = 2 \cdot e^t$, where $t$ is the…
An investor analyzes the growth of a bacteria culture used to manufacture bioplastics. The population in millions follows the formula $P = 2 \cdot e^t$, where $t$ is the time in hours and $e$ is Euler's irrational number ($e \approx 2.71828$).
Calculate the estimated population at $3$ hours using the value of $e$ from your calculator. If by mistake the laboratory used the approximation $e = 2.7$, how many bacteria difference (in millions) will there be compared to the more precise calculation?
Multi-step problems with exact values
Harder problems chain several ideas together. A plan that works:
-
Draw a picture and label what you know.
-
Find a right triangle or a formula that connects the pieces.
-
Do the algebra exactly and simplify radicals using $\sqrt{ab} = \sqrt{a}\sqrt{b}$.
-
Approximate at the end and answer in context, with units.
Two moves come up often. If a square has area $A$, its side is $\sqrt{A}$. If a circle has area $A$, then $\pi r^2 = A$, so $r = \sqrt{A/\pi}$.
In the design of a logo, an equilateral triangle inscribed in a circle of radius $R = 10$ in is used, as shown in the figure. Knowing that in an equilateral triangle the…
In the design of a logo, an equilateral triangle inscribed in a circle of radius $R = 10$ in is used, as shown in the figure.
Knowing that in an equilateral triangle the radius $R$ is twice the apothem $a$, calculate the exact value of the apothem $a$. Then, use the Pythagorean theorem to find the exact length of the side $L$. Show that the perimeter of the triangle is an irrational number.
A circular plot of area $100$ ft$^2$ and a square plot that also has an area of $100$ ft$^2$ are to be fenced. a) Calculate the exact perimeter of both plots. (Remember…
A circular plot of area $100$ ft$^2$ and a square plot that also has an area of $100$ ft$^2$ are to be fenced.
- a) Calculate the exact perimeter of both plots. (Remember that the area of a circle is $A = \pi r^2$).
- b) If the fence costs $\$15$ per foot, which plot will be cheaper to fence? Use all the decimal places on your calculator to give the exact cost in dollars for each one.
Knowing that the area of a circle is $A = \pi \cdot r^2$, imagine that we want to construct a square that has exactly the same area as a circle of radius $10$ in. (This…
Knowing that the area of a circle is $A = \pi \cdot r^2$, imagine that we want to construct a square that has exactly the same area as a circle of radius $10$ in. (This is the ancient mathematical problem of squaring the circle).
Find the exact expression for the length of the side of that square. Is it possible to measure that side exactly in the real world with a ruler graduated in inches? Justify your answer.
Back to the start: what is the exact value of $\sqrt{2}$? Its exact value is the symbol $\sqrt{2}$ itself, the length of the diagonal of a $1$ by $1$ square. Any decimal like $1.4142$ is only an approximation, which is why Emma's check never reached exactly $2$.