Rational number

In mathematics, a rational number is a number that can be written as a fraction. The set of rational number is often represented by the symbol ${\displaystyle \mathbb {Q} }$, standing for "quotient" in English.^{[1] [2]}

Rational numbers are all real numbers, and can be positive or negative. A number that is not rational is called irrational.^[3]

Most of the numbers that people use in everyday life are rational. These include fractions, integers and numbers with finite decimal digits. In general, a number that can be written as a fraction while it is in its own form is rational.

All rational numbers can be written as a fraction. Take 1.5 as an example, this can be written as ${\displaystyle 1{\frac {1}{2}}}$, ${\displaystyle {\frac {3}{2}}}$, or ${\displaystyle 3/2}$.

More examples of fractions that are rational numbers include ${\displaystyle {\frac {1}{7}}}$, ${\displaystyle {\frac {-8}{9}}}$, and ${\displaystyle {\frac {2}{5}}}$.

A terminating decimal is a decimal with a certain number of digits to the right of the decimal point. Examples include 3.2, 4.075, and -300.12002. All of these are rational. Another good example would be 0.9582938472938498234.

A repeating decimal is a decimal where there are infinitely many digits to the right of the decimal point, but which follow a repeating pattern.

An example of this is ${\displaystyle {\frac {1}{3}}}$. As a decimal, it is written as 0.3333333333... The dots indicate that the digit 3 repeats forever.

Sometimes, a group of digits repeats. An example is ${\displaystyle {\frac {1}{11}}}$. As a decimal, it is written as 0.09090909... In this example, the group of digits 09 repeats.

Also, sometimes the digits repeat after another group of digits. An example is ${\displaystyle {\frac {1}{6}}}$. It is written as 0.16666666... In this example, the digit 6 repeats, following the digit 1.

If you try this on your calculator, sometimes it may make a rounding error at the end. For instance, your calculator may say that ${\displaystyle {\frac {2}{3}}=0.6666667}$, even though there is no 7. It rounds the 6 at the end up to 7.

The digits after the decimal point in an irrational number do not repeat in an infinite pattern. For instance, the first several digits of π (Pi) are 3.1415926535... A few of the digits repeat, but they never start repeating in an infinite pattern, no matter how far you go to the right of the decimal point.

• Whenever you add or subtract two rational numbers, you always get another rational number.

• Whenever you multiply two rational numbers, you always get another rational number.

• Whenever you divide two rational numbers, you always get another rational number (as long as you do not divide by zero).

• Two rational numbers ${\displaystyle {\frac {a}{b}}}$ and ${\displaystyle {\frac {c}{d}}}$ are equal if ${\displaystyle ad=bc}$.

• Decimal
• Fraction (mathematics)
• Irrational number
• Real number

1. ↑ "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-11.
2. ↑ "Rational number". Encyclopedia Britannica. Retrieved 2020-08-11.
3. ↑ Weisstein, Eric W. "Rational Number". mathworld.wolfram.com. Retrieved 2020-08-11.