How to Compare Rational Numbers

Rational numbers are any numbers that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, with the denominator \(q\) not being equal to zero. Rational numbers include fractions like \(\frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{7}{4}\), etc., as well as integers like \(1, 2, 3, -1, -2, -3\), and so forth because any integer can be expressed as a fraction with a denominator of \(1\). Rational numbers also include decimal numbers that either terminate (like \(0.75\) or \(0.2\)) or recur (like \(0.3333...\) or \(0.123123123...\)).

How to Compare Rational Numbers

A Step-by-step Guide to Comparing Rational Numbers

Here’s a simple step-by-step process:

Step 1: Understand What Rational Numbers Are

A rational number is a number that can be expressed as the quotient (or fraction) of two integers, where the denominator is not zero. Some examples of rational numbers are \(\frac{1}{2}, \frac{2}{3}, 5, 0, -\frac{3}{4}\), etc.

Step 2: Put Rational Numbers into the Same Format

To compare rational numbers, it’s easiest if they’re in the same format. This means either converting both numbers into the decimal form or expressing both numbers as fractions with the same denominator.

If the numbers are already in fraction form with the same denominator, you can skip this step. If they are not, however, you will need to find a common denominator. This is typically the least common multiple (LCM) of the two denominators.

For example, if you want to compare \(\frac{1}{2}\) and \(\frac{3}{4}\), the common denominator would be \(4\). You can convert \(\frac{1}{2}\) to \(\frac{2}{4}\), and then easily compare \(\frac{2}{4}\) and \(\frac{3}{4}\).

Step 3: Compare the Numerators or Decimals

Once your rational numbers are either in decimal form or expressed as fractions with the same denominator, you can compare them.

If they’re in decimal form, compare them as you would any other number. The number with the greater digits is the larger number.

If they’re expressed as fractions with the same denominator, compare the numerators. The fraction with the larger numerator is the greater fraction.

Continuing the previous example, once you’ve converted \(\frac{1}{2}\) to \(\frac{2}{4}\), you can compare it to \(\frac{3}{4}\) by comparing the numerators \(2\) and \(3\). Since \(3 > 2, \frac{3}{4}\) is greater than \(\frac{1}{2}\).

Step 4: Note the Results

Write down or remember your results. The number that is greater is the larger number when comparing, and the other number is the smaller one.

Step 5: Confirm Your Results

As a final check, you can confirm your results by converting both numbers to their decimal forms and comparing them. If you get the same results as your earlier comparison, you can be confident that your answer is correct.

Keep in mind that if your rational numbers are negative, the number that is “greater” in absolute value is actually the smaller number. For example, \(-\frac{3}{4}\) is less than \(-\frac{1}{2}\).

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