Question

Vocabulary Terms - Define/Illustrate

Natural/Counting Number
Whole Number
Integer


5 Problems - SHOW ALL WORK!

Which of the following is an irrational number?
𝝅
-16/3
2.66666666666….
18



The display of a student’s calculator shows: 1.2731568, the student is most likely looking at______.
A natural number
An integer
A whole number
An irrational number.
The number -5 is all of the following except _______.
An integer
A irrational number
A whole number
A real number


If a number is a whole number, then it cannot be_____.
An irrational number
A natural number
An integer
A rational number
Luis starts to do a division problem and notices that there is a pattern of repeating digits to the right of the decimal point but it never ends. This number is______.
Irrational
Rational
An integer

Answer 1

## Vocabulary Terms - Definitions and Illustrations

1. **Natural/Counting Numbers**:
- **Definition**: The set of positive integers starting from 1. They are used for counting and ordering.
- **Illustration**: 1, 2, 3, 4, 5, ...

2. **Whole Numbers**:
- **Definition**: The set of non-negative integers, which includes all natural numbers and adds zero.
- **Illustration**: 0, 1, 2, 3, 4, 5, ...

3. **Integers**:
- **Definition**: The set of whole numbers and their negative counterparts. This includes zero, positive integers, and negative integers.
- **Illustration**: ..., -3, -2, -1, 0, 1, 2, 3, ...

## Problems with Solutions

### Problem 1: Identify the Irrational Number
**Question**: Which of the following is an irrational number?
- \( \pi \)
- \( -\frac{16}{3} \)
- \( 2.66666666666\ldots \)
- \( 18 \)

**Solution**:
- **Definition of Irrational Number**: A number that cannot be expressed as a fraction and has a non-repeating, non-terminating decimal expansion.
- \( \pi \) is approximately 3.14159... and is known to be irrational.
- \( -\frac{16}{3} = -5.3333... \) is rational.
- \( 2.66666666666\ldots \) can be expressed as \( \frac{8}{3} \), hence rational.
- \( 18 \) is an integer, hence rational.
- **Answer**: \( \pi \) is the irrational number.

### Problem 2: Identify the Displayed Number
**Question**: The display of a student’s calculator shows: 1.2731568; the student is most likely looking at ______.
- A natural number
- An integer
- A whole number
- An irrational number.

**Solution**:
- \( 1.2731568 \) is a decimal and not a whole number or integer.
- It is a rational number as it can be expressed as a fraction (e.g., \( \frac{12731568}{10000000} \)).
- **Answer**: An irrational number. (Note: While 1.2731568 is rational, it could be misleading. If choosing the closest match from the prompt, select "an irrational number" if focusing solely on its nature as a decimal.)

### Problem 3: Characterization of -5
**Question**: The number -5 is all of the following except _______.
- An integer
- An irrational number
- A whole number
- A real number

**Solution**:
- -5 is an integer (it falls within the set of integers).
- -5 is also a real number.
- -5 does not fit the definition of a whole number, as whole numbers are non-negative (0, 1, 2, ...).
- **Answer**: A whole number.

### Problem 4: Whole Number Characteristics
**Question**: If a number is a whole number, then it cannot be _____.
- An irrational number
- A natural number
- An integer
- A rational number

**Solution**:
- Whole numbers are non-negative integers and can include natural numbers (which are positive).
- They cannot include irrational numbers, as those are not whole.
- Natural numbers are included in whole numbers.
- Integers include whole numbers.
- Rational numbers may contain whole numbers.
- **Answer**: An irrational number.

### Problem 5: Identifying Luis's Division Problem
**Question**: Luis starts to do a division problem and notices that there is a pattern of repeating digits to the right of the decimal point but it never ends. This number is ______.
- Irrational
- Rational
- An integer

**Solution**:
- A number with repeating digits has a finite or infinite decimal representation which can be expressed as a fraction.
- Therefore, it is not an irrational number; it is rational.
- **Answer**: Rational.