Question
label each group of numbers as rational,whole,integers,irrational and/or natural.
Group A: 2^2 pi, 8^2
Group B: 11/9 15/5 1.7
Group C {..., -3,-2,-1,0,1,...}
Group D{0,1,2,3,...}
Group E {1,2,3,...}
Group A: 2^2 pi, 8^2
Group B: 11/9 15/5 1.7
Group C {..., -3,-2,-1,0,1,...}
Group D{0,1,2,3,...}
Group E {1,2,3,...}
Answers
GPT-4o mini
Let's categorize each group of numbers based on their properties:
**Group A: \(2^2 \pi, 8^2\)**
- **\(2^2 \pi\)**: This is a product of a rational number \(2^2 = 4\) and the irrational number \(\pi\). Since the product of a rational number and an irrational number is irrational, \(2^2 \pi\) is **irrational**.
- **\(8^2\)**: This equals \(64\), which is a whole number and an integer. So \(8^2\) is **rational**, **whole**, and **integer**.
**Label for Group A**: **Irrational (for \(2^2 \pi\)), Rational, Whole, Integer (for \(8^2\))**
---
**Group B: \(\frac{11}{9}, \frac{15}{5}, 1.7\)**
- **\(\frac{11}{9}\)**: This is a fraction, hence it is **rational**.
- **\(\frac{15}{5}\)**: This simplifies to \(3\), which is a whole number and an integer, so it is also **rational, whole, and integer**.
- **\(1.7\)**: This is a decimal number that can be expressed as \(\frac{17}{10}\), making it **rational**.
**Label for Group B**: **Rational (for all numbers)**, specifically:
- \(\frac{11}{9}\): Rational
- \(\frac{15}{5}\): Rational, Whole, Integer
- \(1.7\): Rational
---
**Group C: \(\{..., -3, -2, -1, 0, 1, ...\}\)**
This set includes all negative whole numbers, zero, and positive whole numbers, which defines the set of integers. Thus, this group is:
- **Integers**
**Label for Group C**: **Integers**
---
**Group D: \(\{0, 1, 2, 3, ...\}\)**
This set includes all non-negative whole numbers starting from zero. This set is defined as:
- **Whole Numbers (or Whole)**
**Label for Group D**: **Whole Numbers**
---
**Group E: \(\{1, 2, 3, ...\}\)**
This set includes all positive whole numbers. Thus, this group is defined as:
- **Natural Numbers**
**Label for Group E**: **Natural Numbers**
**Summary of Group Labels**:
- Group A: Irrational (for \(2^2 \pi\)), Rational, Whole, Integer (for \(8^2\))
- Group B: Rational
- Group C: Integers
- Group D: Whole Numbers
- Group E: Natural Numbers
**Group A: \(2^2 \pi, 8^2\)**
- **\(2^2 \pi\)**: This is a product of a rational number \(2^2 = 4\) and the irrational number \(\pi\). Since the product of a rational number and an irrational number is irrational, \(2^2 \pi\) is **irrational**.
- **\(8^2\)**: This equals \(64\), which is a whole number and an integer. So \(8^2\) is **rational**, **whole**, and **integer**.
**Label for Group A**: **Irrational (for \(2^2 \pi\)), Rational, Whole, Integer (for \(8^2\))**
---
**Group B: \(\frac{11}{9}, \frac{15}{5}, 1.7\)**
- **\(\frac{11}{9}\)**: This is a fraction, hence it is **rational**.
- **\(\frac{15}{5}\)**: This simplifies to \(3\), which is a whole number and an integer, so it is also **rational, whole, and integer**.
- **\(1.7\)**: This is a decimal number that can be expressed as \(\frac{17}{10}\), making it **rational**.
**Label for Group B**: **Rational (for all numbers)**, specifically:
- \(\frac{11}{9}\): Rational
- \(\frac{15}{5}\): Rational, Whole, Integer
- \(1.7\): Rational
---
**Group C: \(\{..., -3, -2, -1, 0, 1, ...\}\)**
This set includes all negative whole numbers, zero, and positive whole numbers, which defines the set of integers. Thus, this group is:
- **Integers**
**Label for Group C**: **Integers**
---
**Group D: \(\{0, 1, 2, 3, ...\}\)**
This set includes all non-negative whole numbers starting from zero. This set is defined as:
- **Whole Numbers (or Whole)**
**Label for Group D**: **Whole Numbers**
---
**Group E: \(\{1, 2, 3, ...\}\)**
This set includes all positive whole numbers. Thus, this group is defined as:
- **Natural Numbers**
**Label for Group E**: **Natural Numbers**
**Summary of Group Labels**:
- Group A: Irrational (for \(2^2 \pi\)), Rational, Whole, Integer (for \(8^2\))
- Group B: Rational
- Group C: Integers
- Group D: Whole Numbers
- Group E: Natural Numbers