Problem 107 - Elementary and Intermediate Algebra

Question

True or false: Every real number can be expressed as a decimal.

Step-by-Step Solution

Answer

True: Every real number can be expressed as a decimal.

Step 1: Understanding Real Numbers

Real numbers include all rational and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot be exactly expressed as fractions.

Step 2: Expressing Rational Numbers as Decimals

Rational numbers are numbers that can be expressed as a ratio of two integers. They can also be written as either terminating or repeating decimals. For example, \( \frac{1}{2} = 0.5 \) (a terminating decimal) and \( \frac{1}{3} = 0.333\ldots \) (a repeating decimal).

Step 3: Expressing Irrational Numbers as Decimals

Irrational numbers cannot be expressed as simple fractions, but they can be represented as non-repeating, non-terminating decimals. An example is \( \pi \), approximately 3.14159, which continues infinitely without repeating.

Step 4: Conclusion

Since both rational and irrational numbers, which together form the real numbers, can be represented as decimals, we conclude that every real number can indeed be expressed as a decimal.

Key Concepts

Rational Numbers

Rational numbers are those numbers that you can express as the ratio of two integers. Simply put, they can be written as a fraction where the numerator is an integer and the denominator is a non-zero integer. This characteristic makes rational numbers quite versatile.
For example, the number \( \frac{3}{4} \) is rational because it can be expressed as the fraction of two integers. When you convert \( \frac{3}{4} \) into a decimal form, you get 0.75, which is a terminating decimal.
But not all rational numbers have this neat ending. Consider \( \frac{1}{3} \); when converted to a decimal, it becomes 0.333... and the threes go on forever. This is called a repeating decimal.

• Rational numbers can be represented as fractions.
• They convert to either terminating or repeating decimals.
• Examples include integers like 5 (which is \( \frac{5}{1} \)), terminating decimals like 0.25, and repeating decimals like 0.6666….

Irrational Numbers

Irrational numbers might seem puzzling because they cannot be expressed as exact fractions. These numbers cannot be written as a neat ratio of two integers. Instead, they are best represented in decimal form as non-terminating and non-repeating decimals.
A common example of an irrational number is \( \pi \). While we often use the approximation 3.14 for \( \pi \), it actually goes on forever without repeating, represented as 3.14159... and beyond.
Another famous irrational number is \( \sqrt{2} \), which also can't be truly captured by any fractional expression. Its decimal is approximately 1.414213..., and it, too, extends infinitely without forming a repeating pattern.

• Irrational numbers cannot be expressed as fractions.
• They are expressed as non-repeating, non-terminating decimals.
• Examples include \( \pi \) and \( \sqrt{2} \).

Decimals

Decimals are a way to express real numbers without using fractions. They are often found on calculators and in everyday financial calculations. Decimals are divided into different types based on their patterns.

• **Terminating Decimals:** These are decimals that come to an end. For instance, 0.5 or 2.75 are terminating because they do not continue past the decimal place.
• **Repeating Decimals:** These are decimals that have a repeating pattern, such as 0.333... or 0.666... Often, these are written with a line over the repeating part for simplicity, like 0.\overline{3}.
• **Non-Repeating, Non-Terminating Decimals:** These decimals go on forever without any repeating pattern. They include the decimals derived from irrational numbers, like \( \pi \), which is approximately 3.14159....

Understanding decimal representation allows us to see real numbers in a form that's easily usable in calculations and interpretations.