Lesson 1: Rational Numbers

Property

To convert a rational number a/b (where a and b are integers) to a decimal, divide the numerator by the denominator using long division. The resulting decimal will either be a terminating decimal (if the division eventually results in a remainder of 0) or a repeating decimal (if a non-zero remainder repeats, creating an infinitely repeating sequence of digits in the quotient).

Examples

• Terminating: To convert 5/8 to a decimal, calculate 5 divided by 8. The division ends with a remainder of 0, resulting in the terminating decimal 0.625.
• Repeating: To convert 2/11 to a decimal, calculate 2 divided by 11. The remainders 2 and 9 alternate endlessly, resulting in the repeating decimal 0.181818...

Explanation

Think of a fraction as a division problem waiting to be solved. When you perform long division, you are looking at the leftovers (remainders). If you eventually have no leftovers, the decimal stops cleanly. If you start seeing the same leftovers over and over, you are caught in a mathematical loop, which means your decimal will repeat that pattern forever.

Property

A terminating decimal is another representation of a fraction whose denominator is not given explicitly, but is understood to be an integer power of ten. A terminating decimal leads to a fraction whose denominator is a product of 2’s and/or 5’s, and conversely, any such fraction is represented by a terminating decimal.

Examples

• To convert $0.425$ to a fraction, we write it as $\frac{425}{1000}$. This simplifies to $\frac{17}{40}$. The denominator $40 = 2^3 \cdot 5$, containing only factors of 2 and 5.
• The fraction $\frac{7}{20}$ has a terminating decimal because its denominator is $20 = 2^2 \cdot 5$. We can write it as $\frac{7 \cdot 5}{20 \cdot 5} = \frac{35}{100} = 0.35$.
• The fraction $\frac{1}{12}$ will not have a terminating decimal because its denominator $12 = 2^2 \cdot 3$ contains a prime factor of 3.

Explanation

If a fraction’s denominator has only prime factors of 2 and 5, you can multiply it to become a power of 10 (like 100 or 1000). This allows it to be written as a decimal that stops, or terminates.