Problem 14 - Elementary Algebra

Question

Determine whether the following real numbers are integers, rational, or irrational. $$ 1.001 $$

Step-by-Step Solution

Answer

1.001 is a rational number.

Step 1: Understanding the Number

The number given is 1.001. It is a decimal number with three digits after the decimal point.

Step 2: Check if It's an Integer

To be an integer, a number must not have any decimal or fractional part. Since 1.001 has a decimal part (0.001), it is not an integer.

Step 3: Check if It's Rational

A rational number is a number that can be expressed as a fraction \( \frac{a}{b} \) where \( a \) and \( b \) are integers, and \( b eq 0 \). The decimal 1.001 can be expressed as the fraction \( \frac{1001}{1000} \). Since this is a fraction with integers, 1.001 is a rational number.

Step 4: Determine if it's Irrational

Irrational numbers cannot be expressed as a fraction of two integers. Since 1.001 can be expressed as a fraction, it is not an irrational number.

Key Concepts

Integers

Integers are the simplest type of rational numbers. They include all the whole numbers you already know, like 0, 1, 2, and so on, as well as their negative counterparts like -1, -2, etc.

What makes integers special is that they do not have fractions or decimal parts. They are complete and whole.

For example:

• 5 is an integer because it has no decimal or fraction.
• -3 is an integer because it is a whole number, just negative.

The number 1.001 was checked against the definition of integers. Because it includes a decimal part (0.001), it is not an integer. This distinction is important in math to differentiate whole numbers from numbers with parts.

Irrational Numbers

Irrational numbers are fascinating because they cannot be expressed as simple fractions. They go on forever after the decimal point without repeating any pattern.

Some of the most well-known irrational numbers are:

• \( \pi \) which is approximately 3.14159...
• \( \sqrt{2} \) which is roughly 1.41421...

These numbers have endless, non-repeating decimals, distinguishing them from rational numbers.

When evaluating whether 1.001 is an irrational number, it was determined it is not. It can be expressed as a fraction, namely \( \frac{1001}{1000} \), meaning it is rational, not irrational. This is because irrational numbers cannot be written as a fraction of two integers.

Decimal Representation

Decimal representation is a way to express numbers using a point to separate the whole number part from the fractional part.

In the number 1.001:

• The whole number part is 1.
• The fractional part is 0.001, which is three decimal places.

This way of writing numbers is useful in everyday life because it gives a clear idea of how big or small the number is beyond whole numbers.

Decimal representation becomes crucial in determining the nature of a number. For instance, knowing the decimal form of 1.001 helped us conclude it was rational because we could convert it to a fraction \( \frac{1001}{1000} \). It's important to recognize how decimals are arranged to identify if the number is whole, rational, or irrational.