Comparing decimals is very easy, but also very important. In fact, it’s the first thing we should learn to do in order to use real numbers (or decimal numbers) properly. So, how do we compare decimals? If numbers to the left of the decimal point of two real numbers are the same (for example: $12.432$ and $12.586$), how to decide which real number is greater than the other?
“In the most commonly used base-$10$ positional system, the value of a position in a number is always smaller than the value of the positions to its left. Also, the value of a position in a number is always larger than the value of the positions to its right.”
Those statements are true no matter which numerals are in those positions, or which side of the decimal point these positions are.
So, when you compare the decimals of two numbers, it makes sense to only compare the decimals on the same position. In our previous example ($12.432$ and $12.586$), we’d compare $4$ to $5$ (both represent the number of tenths), $3$ to $8$ (both represent the number of hundredths), and so on.
We start from the leftmost decimal (the one next to the decimal point – the tenth), and we continue until we find a difference. In our example, the difference is in the first numeral behind the decimal point. Since $4$ is smaller than $5$, we can safely say that $12.586$ is greater than $12.432$, and stop the comparison there. Now, let’s try and practice this a little bit more.
For example, let’s compare these two numbers:
$343.45667$ and $343.45567$
We go through the digits one by one, and see that the first five digits are the same. But the sixth digit of the number $343.45667$ is greater than the sixth digit in number $343.45567$, which means that $A$ is greater than $B$. Now, let’s do a couple more to really get the hang of this. 
Example 2:
When we compare positive and negative real numbers, the rules are the same as if we are comparing positive and negative integers. Therefore, positive real number is always greater than negative one.
Example 3:
You might think these two numbers are the same at first, but watch the position of the decimal point carefully.
Example 4:
We compare two or more negative real numbers in the same way we compare the positive ones. Just remember to reverse the result due to the minuses.
Decimals on the number line
Real numbers can be divided into positive and negative real numbers, as well.
Positive real numbers are found to the right side of the point of origin, and the negative ones are on the left. Between any two whole numbers or, as they are also called – integers, lies an ‘infinitely large number of decimal numbers.
The safest way to be precise about placing a decimal number on the number line is to convert it into a fraction, and then place it on the number line. Let’s do some examples together to really understand how it’s done.
Example:
Let’s place the number $0.25$ on the number line.
As we’ve already learned, we can transform this number into a fraction:
This means that this point is away from the point of origin, and to the right. We’ll divide our segment from $0$ to $1$ into four parts and the first dot marks the position of our number.
Comparing fractions
To compare fractions, you can use same procedure stated above. You can also convert fractions to decimal numbers if you feel more comfortable. However, there are a few things about fractions that are worth mentioning. When the numerators’ value is increasing, the value of the fraction is also increasing. When the denominators’ value is increasing, the value of the fraction is decreasing. When a fraction is negative, the procedure is reversed.
Example:
Compare: $\frac{2}{3}$ and $\frac{3}{4}$.
$\frac{2}{3}$ ? $\frac{3}{4}$
$\frac{a}{b} ? \frac{c}{d}$
$ad$ ? $cd$
In our exapmple $a=2, b=3, c=3, d=4$
$8<9$
Thats means:
$\frac{2}{3}$ < $\frac{3}{4}$
Example:
Compare: $\frac{2}{5}$ and $\frac{4}{7}$.
$\frac{2}{5}$ ? $\frac{4}{7}$.
$14 <20$
$\frac{2}{5} < \frac{4}{7}$
