Problem Analysis:
The task involves classifying numbers into different categories of rational numbers and placing them in the correct location on a Venn diagram. The categories include
Whole Numbers,
Integers,
Rational Numbers, and
Irrational Numbers. Additionally, there are true/false questions to answer.
Step-by-Step Solution:
####
1. Classify Each Number:
We need to determine whether each number is a
Whole Number,
Integer,
Rational Number, or
Irrational Number. Let's analyze each number:
1.
$\sqrt{7}$:
- $\sqrt{7}$ is an irrational number because it cannot be expressed as a fraction of two integers, and its decimal representation is non-repeating and non-terminating.
-
Category: Irrational Numbers
2.
$14.2$:
- $14.2$ can be written as $\frac{142}{10}$, which is a fraction of two integers.
-
Category: Rational Numbers
3.
$8$:
- $8$ is a whole number, an integer, and a rational number.
-
Category: Whole Numbers, Integers, Rational Numbers
4.
$\sqrt{6}$:
- $\sqrt{6}$ is an irrational number because it cannot be expressed as a fraction of two integers, and its decimal representation is non-repeating and non-terminating.
-
Category: Irrational Numbers
5.
$\frac{5}{8}$:
- $\frac{5}{8}$ is a fraction of two integers.
-
Category: Rational Numbers
6.
$0$:
- $0$ is a whole number, an integer, and a rational number.
-
Category: Whole Numbers, Integers, Rational Numbers
7.
$0.32$:
- $0.32$ can be written as $\frac{32}{100} = \frac{8}{25}$, which is a fraction of two integers.
-
Category: Rational Numbers
8.
$7.66$:
- $7.66$ can be written as $\frac{766}{100} = \frac{383}{50}$, which is a fraction of two integers.
-
Category: Rational Numbers
9.
$2\frac{5}{7}$:
- $2\frac{5}{7}$ is a mixed number, which can be converted to an improper fraction: $2\frac{5}{7} = \frac{19}{7}$.
-
Category: Rational Numbers
####
2. Place Numbers on the Venn Diagram:
Based on the classifications above, we can place each number in the appropriate section of the Venn diagram:
-
Whole Numbers: $8$, $0$
-
Integers: $8$, $0$
-
Rational Numbers: $14.2$, $8$, $\frac{5}{8}$, $0$, $0.32$, $7.66$, $2\frac{5}{7}$
-
Irrational Numbers: $\sqrt{7}$, $\sqrt{6}$
####
3. Answer True/False Questions:
11.
All whole numbers are rational numbers.
-
True: Every whole number can be expressed as a fraction with a denominator of 1 (e.g., $8 = \frac{8}{1}$). Therefore, all whole numbers are rational numbers.
12.
$\frac{2}{5}$ is a rational number.
-
True: $\frac{2}{5}$ is already in the form of a fraction of two integers, so it is a rational number.
Final Answer:
1. $\sqrt{7}$: Irrational Numbers
2. $14.2$: Rational Numbers
3. $8$: Whole Numbers, Integers, Rational Numbers
4. $\sqrt{6}$: Irrational Numbers
5. $\frac{5}{8}$: Rational Numbers
6. $0$: Whole Numbers, Integers, Rational Numbers
7. $0.32$: Rational Numbers
8. $7.66$: Rational Numbers
9. $2\frac{5}{7}$: Rational Numbers
11.
True: All whole numbers are rational numbers.
12.
True: $\frac{2}{5}$ is a rational number.
$$
\boxed{\text{See detailed explanations above.}}
$$
Parent Tip: Review the logic above to help your child master the concept of identifying real numbers worksheet.
