Problem 19 - Algebra 2

Question

Name the sets of numbers to which each number belongs. $$ \frac{12}{2} $$

Step-by-Step Solution

Answer

The number 6 belongs to natural, whole, integer, rational, and real number sets.

Step 1: Simplify the Fraction

First, simplify the fraction \( \frac{12}{2} \). Divide the numerator (12) by the denominator (2): \[ \frac{12}{2} = 6 \]. The simplified number is 6.

Step 2: Identify the Set of Natural Numbers

Natural numbers are all positive integers starting from 1. The number 6 is a positive integer greater than 0, so it belongs to the set of natural numbers.

Step 3: Identify the Set of Whole Numbers

Whole numbers include all natural numbers and zero. Since 6 is a natural number and greater than zero, it belongs to the set of whole numbers.

Step 4: Identify the Set of Integers

Integers include all whole numbers and their negative counterparts. Since 6 is a whole number, it is also an integer.

Step 5: Identify the Set of Rational Numbers

Rational numbers can be expressed as a fraction of two integers, where the denominator is not zero. The number 6 can be expressed as \( \frac{6}{1} \), thus it is a rational number.

Step 6: Examine the Set of Irrational Numbers

Irrational numbers cannot be expressed as a simple fraction involving two integers. Since 6 can be expressed as a fraction (\( \frac{6}{1} \)), it is not an irrational number.

Step 7: Examine the Set of Real Numbers

Real numbers include all rational and irrational numbers. Since 6 is a rational number, it is also a real number.

Key Concepts

Natural Numbers

Natural numbers are the counting numbers we use in everyday life. These numbers start from 1 and include all positive integers that follow. They are simple and straightforward.
- Examples: 1, 2, 3, 4...
The key characteristic of natural numbers is that they do not include zero or any negative numbers. So, if you are counting objects, like apples or books, you are using natural numbers.

In our original exercise, when we simplified \( \frac{12}{2} \) to get 6, we found a natural number. This is because 6 is positive and greater than zero, fitting perfectly into the set of natural numbers.

Whole Numbers

Whole numbers expand on the set of natural numbers by adding one more important element: zero. So, whole numbers start from zero and include all the natural numbers thereafter.
- Examples: 0, 1, 2, 3, 4...
This means that whole numbers are ideal for situations where starting from nothing or zero is necessary. For example, if you're calculating your current bank balance or counting inventory where zero items is possible, whole numbers are used.

In our exercise, since 6 is a natural number and already includes zero as a possibility, it's also a whole number. All natural numbers automatically fall into the category of whole numbers.

Integers

Integers take the concept of whole numbers a step further by including negative numbers as well. This forms a complete set of numbers that surround zero.
- Examples: ..., -3, -2, -1, 0, 1, 2, 3...
Integers are useful in real-life scenarios where opposites or the concept of loss is involved, like temperatures below zero or owing money.

From our exercise, since 6 is a whole number, it naturally falls into the category of integers. All whole numbers, including zero and the positive numbers, are part of the larger set of integers.

Rational Numbers

Rational numbers are a broader set that include any number that can be expressed as a fraction or a ratio between two integers, where the denominator is not zero.
- Examples: \( \frac{1}{2}, -\frac{4}{3}, 7 \) (which is \( \frac{7}{1} \))
These numbers cover both the set of fractions and whole numbers, making them essential in many areas of math and real-life calculations, like dividing a pizza into smaller portions.

In the given exercise, 6 is a rational number because it can be written as \( \frac{6}{1} \), fulfilling the criteria of a rational number. Therefore, all integers, including whole and natural numbers, can also be considered rational numbers as they can be expressed in fraction form.