Classifying Real Numbers Worksheet - Free Printable

Problem Overview:

The task is to classify the given numbers into the appropriate categories in the Venn diagram of the real number system. The categories are:
1. Real Numbers (all numbers)
2. Rational Numbers (numbers that can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers, and \( q \neq 0 \))
3. Irrational Numbers (numbers that cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions)
4. Integers (whole numbers and their negatives, e.g., \( \ldots, -2, -1, 0, 1, 2, \ldots \))
5. Whole Numbers (non-negative integers, e.g., \( 0, 1, 2, 3, \ldots \))

The numbers to classify are:
\[ \{-6, 2.73, \frac{3}{7}, \sqrt{2}, \sqrt{9}, -100, 0, \pi, 1, -\frac{1}{2}, -3.8, 5.\overline{42}, 8.293017\ldots\} \]

Step-by-Step Solution:

#### 1. Identify Each Number's Category:

- \(-6\):
- Integer (since it is a whole number with a negative sign).
- Rational (since it can be written as \( \frac{-6}{1} \)).
- Real (all integers are real numbers).

- \(2.73\):
- Rational (terminating decimal, can be written as \( \frac{273}{100} \)).
- Real (all rational numbers are real numbers).

- \(\frac{3}{7}\):
- Rational (it is already in fractional form).
- Real (all rational numbers are real numbers).

- \(\sqrt{2}\):
- Irrational (the square root of 2 is a non-repeating, non-terminating decimal).
- Real (all irrational numbers are real numbers).

- \(\sqrt{9}\):
- Simplifies to \(3\) (since \( \sqrt{9} = 3 \)).
- Whole number (since 3 is a non-negative integer).
- Integer (since 3 is an integer).
- Rational (since 3 can be written as \( \frac{3}{1} \)).
- Real (all integers are real numbers).

- \(-100\):
- Integer (negative whole number).
- Rational (since it can be written as \( \frac{-100}{1} \)).
- Real (all integers are real numbers).

- \(0\):
- Whole number (since 0 is non-negative).
- Integer (since 0 is an integer).
- Rational (since 0 can be written as \( \frac{0}{1} \)).
- Real (all integers are real numbers).

- \(\pi\):
- Irrational (π is a well-known irrational number with a non-repeating, non-terminating decimal expansion).
- Real (all irrational numbers are real numbers).

- \(1\):
- Whole number (since 1 is a non-negative integer).
- Integer (since 1 is an integer).
- Rational (since 1 can be written as \( \frac{1}{1} \)).
- Real (all integers are real numbers).

- \(-\frac{1}{2}\):
- Rational (it is already in fractional form).
- Real (all rational numbers are real numbers).

- \(-3.8\):
- Rational (terminating decimal, can be written as \( \frac{-38}{10} \)).
- Real (all rational numbers are real numbers).

- \(5.\overline{42}\):
- Rational (repeating decimal, which can be expressed as a fraction).
- Real (all rational numbers are real numbers).

- \(8.293017\ldots\):
- Irrational (non-repeating, non-terminating decimal).
- Real (all irrational numbers are real numbers).

#### 2. Place Numbers in the Venn Diagram:

- Whole Numbers: \(0, 1, \sqrt{9}\) (which simplifies to 3).
- Integers: \(-6, -100, 0, 1, \sqrt{9}\) (which simplifies to 3).
- Rational Numbers: \(-6, 2.73, \frac{3}{7}, -100, 0, 1, -\frac{1}{2}, -3.8, 5.\overline{42}\).
- Irrational Numbers: \(\sqrt{2}, \pi, 8.293017\ldots\).
- Real Numbers: All the numbers listed above.

#### 3. Answer the True/False Questions:

1. All whole numbers are integers.
- True: Whole numbers are a subset of integers. Every whole number is also an integer.

2. All integers are whole numbers.
- False: Integers include negative numbers, but whole numbers do not. For example, \(-6\) is an integer but not a whole number.

3. Some rational numbers are integers.
- True: Integers are a subset of rational numbers. For example, \(1\) and \(-6\) are both integers and rational numbers.

4. Some whole numbers are irrational numbers.
- False: Whole numbers are always rational (they can be written as fractions with a denominator of 1). There are no whole numbers that are irrational.

Final Answer:

\[
\boxed{
\begin{aligned}
&\text{Whole Numbers: } 0, 1, \sqrt{9} \\
&\text{Integers: } -6, -100, 0, 1, \sqrt{9} \\
&\text{Rational Numbers: } -6, 2.73, \frac{3}{7}, -100, 0, 1, -\frac{1}{2}, -3.8, 5.\overline{42} \\
&\text{Irrational Numbers: } \sqrt{2}, \pi, 8.293017\ldots \\
&\text{Real Numbers: All the numbers listed} \\
&\text{True/False:} \\
&1. \text{True} \\
&2. \text{False} \\
&3. \text{True} \\
&4. \text{False}
\end{aligned}
}
\]