Rational & irrational number

A rational number is a number that can be expressed as a fraction or ratio (rational). The numerator and the denominator of the fraction are both integers.
When the fraction is divided out, it becomes a terminating or repeating decimal.
(The repeating decimal portion may be one number or a billion numbers.)
Rational numbers can be ordered on a number line.
Examples of rational numbers are :
6 or can also be written as 6.0
-2 or can also be written as -2.0
can also be written as .5
can also be written as -1.25
can also be written as
can also be written as 0.666666666...

can also be written as 0.38181818...

can also be written as 0.62855421687...
the decimals will repeat
after 41 digits
Be careful when using your calculator to determine if a decimal number is irrational. The calculator may not be displaying enough digits to show you the repeating decimals, as was seen in the last example above.


Examples: Write each rational number as a fraction:
1. 0.3
2. 0.007
3. -5.9

Hint: When checking to see which fraction is larger, change the fractions to decimals by dividing.
Examples: Which of the given numbers is greater?
1. .6666666667 > .25 Using full
calculator
display.
2. -2.333333333 > -3.666666667



An irrational number cannot be expressed as a fraction.
Irrational numbers cannot be represented as terminating or repeating decimals.
Irrational numbers are non-terminating, non-repeating decimals.

Examples of irrational numbers are:
= 3.141592654…..
= 1.414213562…..
and 0.12122122212…


Note: Many students think that is the terminating decimal, 3.14, but it is not. Yes, certain math problems ask you to use as 3.14, but that problem is rounding the value of to make your calculations easier. is actually a non-ending decimal and is an irrational number.

Rational and irrational numbers are real numbers.




Irrational number


An irrational number is a non-repeating, non-terminating decimal. It’s decimal representation, is an approximation of its value. Irrational numbers are rounded when written in decimal form.

We can take advantage of the square root key
on a calculator to find approximations for some irrational numbers.

Example: = 2.236067977…….. Since it is impossible to write out the entire decimal (since it never ends) we may approximate to be 2.2 or 2.24 or 2.236, etc., depending upon the rounding directions given in the problem.
If no specific rounding directions are given in a problem, work with the full calculator display, or work with the number in its original form (in this example, work with .)

You should always work with the "full" value of a number (such as ), or the full calculator display of the number, in a multi-step problem, saving the final rounding for the last step. Don't round too soon!
**** If using a calculator, work with the full calculator entries until you are ready to round your final answer.