7.1 Rational and Irrational Numbers

Understanding Rational and Irrational Numbers

Every number you'll work with in this course falls somewhere on the real number line. The two big categories are rational and irrational numbers. Knowing which type a number belongs to helps you understand how numbers behave and how they relate to each other.

Rational vs. Irrational Numbers

Rational numbers can be written as a fraction $\frac{a}{b}$, where $a$ and $b$ are both integers and $b \neq 0$. This includes more types of numbers than you might expect:

• Integers like $3$ (which is really $\frac{3}{1}$)
• Fractions like $\frac{2}{5}$
• Terminating decimals like $0.75$ (which equals $\frac{3}{4}$)
• Repeating decimals like $0.\overline{3}$ (which equals $\frac{1}{3}$)

The key test: if the decimal either stops or repeats a pattern, it's rational.

Irrational numbers are the opposite. They cannot be written as a fraction of two integers. Their decimals go on forever without ever settling into a repeating pattern. Common examples include $\sqrt{2}$, $\pi$, and $\sqrt{5}$.

A common mistake is thinking that a long decimal must be irrational. That's not true. The decimal $0.333333...$ goes on forever, but it repeats, so it's rational. An irrational number like $\sqrt{2} = 1.41421356...$ never repeats.

Decimal to Fraction Conversions

Terminating decimals to fractions:

1. Write the decimal over a power of 10 that matches the number of decimal places.
2. Simplify the fraction.

For example: $0.25 = \frac{25}{100} = \frac{1}{4}$

Repeating decimals to fractions take a bit more work. Here's the method using $0.\overline{3}$ as an example:

1. Let $x = 0.\overline{3}$

2. Multiply both sides by 10: $10x = 3.\overline{3}$

3. Subtract the original equation from this new one: $10x - x = 3.\overline{3} - 0.\overline{3}$

4. Simplify: $9x = 3$

5. Solve: $x = \frac{1}{3}$

The trick in step 2 is choosing the right power of 10. If one digit repeats, multiply by 10. If two digits repeat (like $0.\overline{36}$), multiply by 100. Match the power of 10 to the length of the repeating block.

Fractions to decimals: Just divide the numerator by the denominator. For example, $\frac{3}{8} = 0.375$.

Categorization of Real Numbers

Real numbers include every number on the number line. They break down into subsets that nest inside each other:

• Whole numbers: non-negative integers ($0, 1, 2, 3, \ldots$)
• Integers: whole numbers plus negatives ($\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$)
• Rational numbers: anything expressible as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$
• Irrational numbers: real numbers that are not rational ($\sqrt{3}$, $\pi$)

Relationships Between Number Sets

These sets nest like boxes inside boxes:

$\text{Whole numbers} \subset \text{Integers} \subset \text{Rational numbers} \subset \text{Real numbers}$

Every whole number is an integer, every integer is rational, and every rational number is real. Irrational numbers are also real, but they sit completely outside the rational numbers. There's no overlap between rational and irrational.

Together, rational and irrational numbers account for every point on the number line. The rational numbers alone leave gaps (where numbers like $\sqrt{2}$ live), and the irrational numbers fill those gaps. Between any two real numbers, no matter how close together, there are infinitely many more real numbers. This property is called density.