Lesson 1: What Is a Rational Number?

Property

A rational number is a number that can be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. All fractions, both positive and negative, are rational numbers.
Since any integer, terminating decimal, or repeating decimal can be written as a ratio of two integers, they are all rational numbers.

Examples

• To write the integer $-25$ as a ratio of two integers, express it as a fraction with a denominator of 1: $\frac{-25}{1}$.
• The decimal $9.37$ can be written as a mixed number $9\frac{37}{100}$, which converts to the improper fraction $\frac{937}{100}$.
• The mixed number $-4\frac{2}{3}$ is equivalent to the improper fraction $-\frac{14}{3}$.

Explanation

Think of 'rational' as 'ratio-nal.' Any number that can be expressed as a simple fraction or ratio between two integers is a rational number. This includes whole numbers, integers, and decimals that either end or repeat predictably.

Property

A decimal number is a number that includes a decimal point, separating the whole number part from the fractional part.
The digits to the right of the decimal point represent fractions with denominators that are powers of ten (tenths, hundredths, thousandths, etc.).
For example, a number $a.bcd$ can be written in expanded form as:

Examples

• The decimal $0.6$ is equivalent to the fraction $\frac{6}{10}$.
• The decimal $5.42$ can be written as $5 + \frac{4}{10} + \frac{2}{100}$, which is equal to the mixed number $5\frac{42}{100}$.
• In money, $1.75$ dollars represents one whole dollar and seventy-five hundredths of a dollar, or $1 + \frac{75}{100}$ dollars.

Explanation

A decimal is a way to write a number that is not whole. The decimal point acts as a separator between the whole part on the left and the fractional part on the right. Each place value to the right of the decimal point is ten times smaller than the place value to its left. Understanding this structure is key to performing arithmetic with decimals, especially in financial contexts like calculating costs and change.

Property

To name a decimal number:

1. Name the number to the left of the decimal point (the whole number).
2. Write 'and' for the decimal point.
3. Name the number to the right of the decimal point as if it were a whole number.
4. Name the decimal place of the last digit. The 'th' at the end of the name means the number is a fraction.

Examples

• The number 4.3 is read as 'four and three tenths' because the 3 is in the tenths place.
• The number 2.45 is read as 'two and forty-five hundredths' because the last digit, 5, is in the hundredths place.
• The number 0.009 is read as 'nine thousandths'. We don't name the zero whole number.

Explanation

Naming decimals is like telling a number's full story. The part before 'and' is the whole number, and the part after is the fraction. The last word, like 'hundredths', tells you the size of the fractional pieces.