Problem 35 - Intermediate Algebra

Question

List the elements of $$ \left\\{-3,-\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75,916 . \overline{6}\right\\} $$ that belong to the following sets. Natural numbers

Step-by-Step Solution

Answer

Natural numbers: 1, 2.

Step 1: Understand Natural Numbers

Natural numbers are the set of positive integers starting from 1. They include numbers like 1, 2, 3, and so on, but do not include zero, negative numbers, fractions, or irrational numbers.

Step 2: Identify Natural Numbers in the Set

Go through each element of the set: \(-3, -\frac{8}{5}, 0, \frac{2}{3}, 1, \sqrt{3}, 2, \pi, 4.75, 916 .\overline{6}\). Check if each element is a positive integer without any fractional or decimal part.

Step 3: List Only Positive Integers

Of the given elements, \(1\) and \(2\) are positive integers. These satisfy the criteria for natural numbers.

Step 4: Conclude Natural Numbers

The only elements from the provided set that are natural numbers are \(1\) and \(2\).

Key Concepts

Positive Integers

Positive integers are numbers that you start counting with automatically. They begin at 1 and go up one number at a time — 1, 2, 3, 4, and so on. These numbers are not only positive, but they also don't have any fractional or decimal parts. This means they are whole numbers that you cannot divide further without going into fractions or decimals.

They are called 'positive' because there’s no negative sign in front of them. They sit right next to zero on the number line but start one spot over. So while zero is neutral, 1 is the first positive integer. Why don't zero, negatives, or fractions count as positive integers? Here’s why:

• Zero is a neutral number, which means it's neither positive nor negative.
• Negative numbers have a minus sign (-) before them, which immediately makes them not positive.
• Fractions and decimals are parts of whole numbers, not complete whole numbers like integers.

This property makes positive integers unique within number sets.

Number Sets

Number sets are like groups that numbers belong to based on their characteristics. Just like how animals are grouped into mammals, birds, reptiles, etc., numbers are grouped too.

There are several sets of numbers, and here's a breakdown of the most common:

• Natural Numbers: These include positive integers starting from 1 upward (1, 2, 3...).
• Whole Numbers: Natural numbers plus zero (0, 1, 2, 3...).
• Integers: All whole numbers, including negative numbers (-3, -2, -1, 0, 1, 2, 3...).
• Rational Numbers: Numbers that can be expressed as a fraction of two integers (like \( \frac{1}{2} \) or 4/3).
• Irrational Numbers: Numbers that cannot be written as a simple fraction (like \( \pi \) or \( \sqrt{2} \)).

Each set has unique properties and serves as a way to understand how different numbers behave and relate to each other. For example, natural and whole numbers are subsets of integers, and natural numbers specifically represent whole counts.

Fractions vs Whole Numbers

Fractions and whole numbers seem like polar opposites because they represent quantities in different ways. A whole number is just as it sounds: a complete thing. There are no parts, portions, or fractions to worry about.

Whole numbers are straightforward and easy to grasp because they complete an entire unit. They can be used to count people, cars, or cookies, and there’s no leftover amount.
Unlike whole numbers:

• Fractions: These represent parts of a whole. The top number, or numerator, shows how many parts we have. The bottom number, or denominator, shows the total number of equal parts something is divided into. For example, \( \frac{3}{4} \) means we have 3 parts out of a total of 4 whole parts.
• Fractions are great for talking about things that are divided or shared, like slicing a pizza.
• They are seen as part of rational numbers but are not considered whole numbers.

In mathematical terms, a whole number cannot be a fraction or decimal. That makes them pure and simple, ideal for counting and ordinary calculations.