Problem 27 - Elementary Algebra

Question

True or false. All terminating decimal numbers are rational.

Step-by-Step Solution

Answer

True, all terminating decimal numbers are rational.

Step 1: Understanding the Definitions

Firstly, recall that a rational number is any number that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \). A terminating decimal is a decimal number that has a finite number of digits after the decimal point.

Step 2: Expressing Terminating Decimals as Fractions

A terminating decimal can always be expressed as a fraction. For example, the decimal 0.25 can be written as \( \frac{25}{100} \) which simplifies to \( \frac{1}{4} \), clearly a rational number. This is true for any terminating decimal.

Step 3: General Case

In general, any terminating decimal can be expressed as a fraction. Suppose a terminating decimal has \( n \) digits: \( d.d_1d_2...d_n \). This can be written as \( \frac{d \times 10^n + d_1 \times 10^{n-1} + ... + d_n \times 10^0}{10^n} \), which is a fraction. Hence, it is rational.

Step 4: Conclusion

Since a terminating decimal can always be expressed in the form of \( \frac{a}{b} \), where both \( a \) and \( b \) are integers, and \( b eq 0 \), all terminating decimals are indeed rational numbers.

Key Concepts

Terminating Decimals

Terminating decimals are numbers that "end" after a certain number of digits. This means that there are no infinite repeating numbers following the decimal point. For example, the number 0.75 is a terminating decimal because it stops after two places beyond the decimal point.

One way to identify a terminating decimal is by looking at its denominator when expressed as a fraction. If, after simplification, the denominator is a power of 10 (like 10, 100, or 1000), then the decimal will terminate. For instance:

• 0.5 can be written as \( \frac{5}{10} \) or \( \frac{1}{2} \), a terminating decimal.
• 0.125 can be written as \( \frac{125}{1000} \), which simplifies to \( \frac{1}{8} \), also a terminating decimal.

Understanding terminating decimals is essential because it helps in knowing how decimals can translate into fractions, and vice versa. Knowing this helps us better understand how numbers are structured in mathematics.

Fractions

Fractions are fundamental in understanding many mathematical concepts. A fraction is a part of a whole and is written with two integers, one above the other, separated by a line: \( \frac{a}{b} \). The number "a" is the numerator, and "b" is the denominator, which tells us how many parts make up a whole.

For example, \( \frac{3}{4} \) implies 3 parts out of 4. It is crucial to know that the denominator should never be zero because division by zero is undefined. Every fraction can be considered as a division operation (\( a \div b \)).

Fractions can express numbers between whole numbers, like \( \frac{1}{2} \) which equals 0.5, a terminating decimal. They can also express whole numbers (like \( \frac{4}{2} = 2 \)). Understanding fractions allows individuals to grasp more complex concepts like rational numbers and decimal conversions.

Mathematics Definitions

Understanding definitions in mathematics is crucial for solving problems accurately. Let's define some important terms:

• Rational Numbers: Any number that can be expressed as a fraction \( \frac{a}{b} \), where "a" and "b" are integers and \( b eq 0 \). This includes whole numbers, fractions, and finite decimals.
• Terminating Decimals: Decimals that end after a certain number of digits. They can be rewritten as fractions, making them a subset of rational numbers.
• Integers: Whole numbers that can be positive, negative, or zero, yet do not include fractions or decimals.

Understanding these definitions uncovers the relationships between different types of numbers, aiding in the comprehension of how mathematical concepts interlink. It helps students see the broader picture of rationality in numbers, revealing that all terminating decimals are rational by nature.