A number is rational if it is equal to a ratio of
two integers: that is, if it can be written as a fraction $$m/n$$
where $m$ and $n$ are integers.
| Is $$3/5$$ a rational number?
| |
Notice that the number 2 can be written as $$2/1$$.
| Find a fraction that is equal to 3.
| |
| Is every integer a rational number?
| |
Remember that we can add and subtract fractions by putting them over a common denominator,
and the result will still be a fraction. Similarly, if you multiply or divide two fractions,
the result will again be a fraction.
| Is $$3/43 + 7/12$$ a rational number?
(You don’t have to compute it.)
| |
| Is $$7/22(355/113)$$ a rational number?
| |
If $q$ and $r$ are rational numbers, then so are $q+r$, $q-r$, and $qr$.
If $r ≠ 0$, $$q/r$$ is also a rational number.
(The symbol ‘$≠$’ means “is not equal to.”)
One common way of writing rational numbers is as repeating
decimals. Remember that when we write a decimal with a bar over some
of its numbers, like $1.\ov{27}$, what we mean is that those numbers are repeated
infinitely:
$$
1.\ov{27}=1.27272727...
$$
You can use algebra to convert any repeating decimal into a fraction. For example, you
can convert $1.\ov{27}$ to a fraction as follows:
$$
\cl"tight"{\table
x, , , =, 1.\ov{27};
x, , , =, 1.27\ov{27}, , ,
\; \cl"hint"\text"pulling out the first repetition";
100x, , , =, 127.\ov{27}, , ,
\; \cl"hint"\text"multiplying both sides by 100";
100x, -, x, =, 127.\ov{27}, -, x,
\; \cl"hint"{\text"subtracting"\,x\,\text"from both sides..."};
\colspan 3 99x, =, 127.\ov{27}, -, 1.\ov{27},
\; \cl"hint"{\text"...and using the fact that"\,x= 1.\ov{27}\,\text"..."};
\colspan 3 99x, =, \colspan 3 126,
\; \cl"hint"\text"...to make the repeated part cancel out";
\colspan 3 {99 x} / 99, =, \colspan 3 126/99;
\colspan 3 x, =, \colspan 3 14/11
}
$$
We multiplied by $100=10^2$ in order to shift the repeating decimal over by 2 decimal places,
because there were 2 repeated digits. In general, if there are $n$ repeated digits, then we
multiply by $10^n$ to shift the repeating decimal over by $n$ decimal places.
Convert the following repeating decimals into fractions.
Irrational numbers
Having seen a large number of examples of rational numbers, you might wonder whether every
number is rational. We’ll look at whether $√2$ is rational.
Finding out whether $√2$ is rational is the same as finding out whether there is
a fraction whose square is 2. Using your calculator or the grid on the left, compute the square
of each of the following fractions as a decimal, rounded to as many decimal places as fit in the
answer box.
| Fraction and its square | Is the square
equal to 2?
|
|---|
| As you can see, the squares of the numbers in the table get very
close to 2. Are any of them exactly equal to 2?
| |
| You can quickly square any rational number by entering it in the input
boxes below the grid to the left. Can you find any rational number whose square is exactly
equal to 2?
| |
In fact, there is no such rational number. If you would like to see a proof of this,
click here.
A number which is not rational is called
irrational.
So $√2$ is an irrational number. In fact:
If $n$ is a positive integer and $√n$ is not an integer, then $√n$
is an irrational number.
This is also proved at the link above.
Arithmetic with irrational numbers
In the last question, we learned that square roots such as $√2$ or $√37$ are irrational.
We’ll now look at more complicated expressions formed from those square roots, such as $√2+1$
or $5√37$.
| We’ll start by trying to figure out if the number $√2+1$
is rational or irrational. What is a simpler way of writing the sum
$(√2+1)+(-1)$?
| |
| Is that sum rational or irrational?
| |
| As you saw in ratDef, the sum of two
rational numbers is rational. If $q$ is a rational number, is the sum $q+(-1)$ rational or
irrational?
| |
| If $q=√2+1$, the sum $q+(-1)$ is irrational. Can $√2+1$ be a rational
number?
| |
| Is $√2+1$ rational or irrational?
| |
| Now, we’ll try to determine whether or not $5√37$ is rational. What is
a simpler way of writing $$1/5(5√37)$$?
| |
| Is that product rational or irrational?
| |
| If $q$ is a rational number, is the product $$1/5 q$$
rational or irrational?
| |
| If $q=5√37$, the product $$1/5 q$$ is irrational. Can
$5√37$ be a rational number?
| |
| Is $5√37$ rational or irrational?
| |
In this way, we can see that:
If $q$ is a rational number and $s$ is an irrational number, then $s+q$
is irrational. If $q ≠ 0$, $qs$ is also irrational.
Now, we’ll look at what happens when you divide a rational number by an irrational
number, by considering the specific example $$2/√3$$.
| We’re trying to figure out if the number
$$2/√3$$ is rational or irrational. What is a simpler way of writing the product
$$2/√3 (√3)$$?
| |
| Is that product rational or irrational?
| |
| If $q$ is a nonzero rational number, is the product $q√3$ rational or
irrational?
(Remember that the product of a nonzero rational number and an irrational
number is irrational.)
| |
| If $$q=2/√3$$, the product $q√3$ is rational. Can
$$q=2/√3$$ be a rational number?
| |
| Is $$q=2/√3$$ rational or irrational?
| |
In this way, we can also see that:
If $q$ is a nonzero rational number and $s$ is an irrational number,
then $$q/s$$ is irrational.