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How Do You Put Real Numbers in Order? | Virtual Nerd

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To convert a Decimal to a Fraction follow these steps:

Step 1: Write down the decimal divided by 1, like this:

Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)

Step 3: Simplify (or reduce) the fraction

Example: Convert 0.75 to a fraction

Step 1: Write down 0.75 divided by 1:

Step 2: Multiply both top and bottom by 100 (because there are 2 digits after the decimal point so that is 10×10=100):

× 100

=

× 100

(Do you see how it turns the top number
into a whole number?)
Step 3: Simplify the fraction (this took me two steps):
Note: 75/100 is called a decimal fraction and 3/4 is called a common fraction !
Example: Convert 0.625 to a fraction

Step 1: write down:

Step 2: multiply both top and bottom by 1,000 (3 digits after the decimal point, so 10×10×10=1,000)

Step 3: Simplify the fraction (it took me two steps here):

When there is a whole number part, put the whole number aside and bring it back at the end:

Example: Convert 2.35 to a fraction
Put the 2 aside and just work on 0.35
Step 1: write down:

Step 2: multiply both top and bottom by 100 (2 digits after the decimal point so that is 10×10=100):

Step 3: Simplify the fraction:

÷ 5

=

÷ 5

Bring back the 2 (to make a mixed fraction):

Example: Convert 0.333 to a fraction

Step 1: Write down:

Step 2: Multiply both top and bottom by 1,000 (3 digits after the decimal point so that is 10×10×10=1,000)

Step 3: Simplify Fraction:

Can't get any simpler!

But a Special Note:

If you really meant 0.333... (in other words 3s repeating forever which is called 3 recurring) then we need to follow a special argument. In that case we write down:

Then multiply both top and bottom by 3:

And 0.999... = 1 (Does it? - see the 9 Recurring discussion for more if you are interested), so:

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Step 1: Write down the decimal divided by 1, like this: Step 2: Multiply both top and bottom by 10 for every number after the decimal point. (For example, if there are two numbers after the decimal point, then use 100, if there are three then use 1000, etc.)

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$$ \sqrt{7} $$

Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{7} $$ .

$$ \sqrt{5} $$

Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{5} $$ .

$$ \pi $$

$$ \pi $$ is probably the most famous irrational number out there!

$$ \frac{ \sqrt{2}}{3} $$

Although this number can be expressed as a fraction, we need more than that, for the number to be rational . The fraction's numerator and denominator must both be integers, and $$\sqrt{2} $$ cannot be expressed as an integer.

$$.2020020002 ...$$

This non terminating decimal does not repeat . So, just like $$ \pi $$, it constantly changes and can not be represented as a quotient of two integers

Rational because it can be written as $$ -\frac{12}{1}$$, a quotient of two integers.

Rational, because you can simplify $$ \sqrt{25} $$ to the integer $$ 5 $$ which of course can be written as $$ \frac{5}{1} $$, a quotient of two integers.

This is irrational, the ellipses mark $$ \color{red}{...} $$ at the end of the number $$ \boxed{ 0.09009000900009 \color{red}{...}} $$, means that the pattern of increasing the number of zeroes continues to increase and that this number never terminates and never repeats.

This is rational. All repeating decimals are rational (see bottom of page for a proof.)

This is irrational. You cannot simplify $$ \sqrt{3} $$ which means that we can not express this number as a quotient of two integers.

Unlike the last problem , this is rational. You can simplify $$ \sqrt{9} \text{ and also } \sqrt{25} $$. If you simplify these square roots, then you end up with $$ \frac{3}{5} $$, which satisfies our definition of a rational number (ie it can be expressed as a quotient of two integers)

This is rational because you can simplify the fraction to be the quotient of two integers (both being the number 1)

$ \frac{ \pi}{\pi } = \frac{ \cancel {\pi} } { \cancel {\pi} } = \frac{1}{1}=1 $

This is rational because you can simplify the fraction to be the quotient of two inters (both being the number 1)

$ \frac{ \sqrt{2}}{\sqrt{2} } = \frac{ \cancel {\sqrt{2}} } { \cancel {\sqrt{2}}} = \frac{1}{1}=1 $

let

$$ x = .\overline{1} $$

multiply both sides by 10

subtract equation 1 from 2

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Rational and irrational numbers explained with examples and non examples

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sqrt{7} $$ Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{7} $$ . $$ \sqrt{5} $$ Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{5} $$ . $$ \pi $$ $$ \pi $$ is probably the most famous irrational number out there!

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A Rational Number can be made by dividing two integers.
(An integer is a number with no fractional part.)

Example:

1.5 is a rational number because 1.5 = 3/2 (3 and 2 are both integers)

Most numbers we use in everyday life are Rational Numbers.

Here are some more examples:

Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are not rational they are called Irrational.

Another famous irrational number is :

Formal Definition of Rational Number

More formally we say:
A rational number is a number that can be in the form p/q
wherep and q are integers and q is not equal to zero.
So, a rational number can be:

Where q is not zero

Examples:

Just remember: q can't be zero

Using Rational Numbers

Fun Facts ....

The ancient greek mathematician Pythagoras believed that all numbers were rational, but one of his students Hippasus proved (using geometry, it is thought) that you could not write the square root of 2 as a fraction, and so it was irrational.

But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!

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A Rational Number can be made by dividing two integers. (An integer is a number with no fractional part.) 1.5 is a rational number because 1.5 = 3/2 (3 and 2 are both integers) Most numbers we use in everyday life are Rational Numbers. Here are some more examples: Oops!

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The Real Numbers had no name before Imaginary Numbers were thought of. They got called "Real" because they were not Imaginary. That is the actual answer!