C1.1; E1.1 - Rational numbers and reciprocals

Rational numbers

A fraction describes a division. In $\frac{3}{4}$, the top number, or numerator, is 3 and the bottom number, or denominator, is 4. It means $3\div4$. A whole number can be written this way too: $5=\frac{5}{1}$.

The fraction need not be positive or less than 1. Its appearance does not decide whether it is rational; an exact integer-fraction representation does.

A terminating decimal ends after finitely many places. Multiplying by a suitable power of 10 writes it over an integer denominator, as the table does for 0.125. A recurring decimal repeats the same digit or block forever; it too is rational. For example, dividing 1 by 3 repeatedly leaves remainder 1, producing the same next digit 3 each time. The dots in $0.333\ldots$ matter: the exact value continues forever, while $0.333$ is the different fraction $\frac{333}{1000}$.

Use the value of each expression rather than its written form.

Irrational numbers

Some exact values cannot be written as a fraction of integers. The number $\sqrt{2}$ lies between 1 and 2, because $1^2<2<2^2$, but no integer fraction gives its exact value. A decimal display can locate it accurately without naming it exactly.

For a positive whole number, its square root is rational exactly when that whole number is a perfect square. Thus $\sqrt{64}=8$ is rational, while $\sqrt{65}$ is irrational. It is not enough to say a number “has lots of decimal places”: recurring decimals also have infinitely many places.

The number $\pi$ is another irrational number. Fractions such as $\frac{22}{7}$, and finite decimals such as 3.142, approximate $\pi$; neither is exactly $\pi$.

Compare the exact root $\sqrt{2}$ with a rounded display:

The symbol $\approx$ means approximately equal. The displayed 1.414 is rational, because it equals $\frac{1414}{1000}$. Its square is $1.999396$, close to 2 but not equal to 2. Rounding an irrational number produces a rational approximation; it does not change the type of the exact original number.

Separate the exact expression from the decimal used to estimate it.

Reciprocals

Multiplying by 4 makes a quantity four times as large. Multiplying that result by $\frac14$ returns the original quantity, because $4\times\frac14=1$. These two numbers undo each other's multiplying effect.

Write a whole number over 1 before interchanging numerator and denominator. Keep the sign: a negative number needs a negative reciprocal to make a positive product of 1. A reciprocal is different from an opposite: the opposite of 4 is $-4$, while its reciprocal is $\frac14$.

Keep the negative sign in the next example, but change from an integer to a fraction. The same interchange now gives a reciprocal whose size exceeds 1.

For a decimal, first express the exact value as a fraction. For example, $0.4=\frac4{10}$, so its reciprocal is $\frac{10}{4}=2.5$. Indeed, $0.4\times2.5=1$. A positive number below 1 has a reciprocal above 1; a positive number above 1 has a reciprocal below 1. The reciprocal of 1 is 1, and that of $-1$ is $-1$.

Express the decimal as a fraction before reversing it.

Quick Recap

• A rational number is an integer fraction with a non-zero denominator.
• Irrational decimals neither terminate nor recur; a non-square positive integer has an irrational square root.
• A finite display approximating an irrational value is rational, while the original exact value remains irrational.
• A reciprocal multiplies its non-zero original number to give 1.

The opening display 1.414 is a rational approximation to $\sqrt2$, not its exact value.