Problem 454 - Prealgebra

Question

Determine which of the following are (a) counting numbers (b) whole numbers. $$0,3,25$$

Step-by-Step Solution

Answer

(a) Counting numbers: 3, 25. (b) Whole numbers: 0, 3, 25.

Step 1 - Understand Counting Numbers

Counting numbers, also known as natural numbers, are the set of positive integers starting from 1, 2, 3, and so on. They do not include zero or any negative numbers.

Step 2 - Understand Whole Numbers

Whole numbers are the set of all non-negative integers. This includes 0, 1, 2, 3, and so forth. They do not include any negative numbers or fractions.

Step 3 - Identify Counting Numbers

From the given list \(0, 3, 25\), identify the counting numbers. Since counting numbers are positive integers starting from 1, the counting numbers in the list are \(3, 25\).

Step 4 - Identify Whole Numbers

From the given list \(0, 3, 25\), identify the whole numbers. Since whole numbers include all non-negative integers, the whole numbers in the list are \(0, 3, 25\).

Key Concepts

Counting Numbers

Counting numbers are those we use when we are counting objects. Imagine you are counting apples - you start at 1, then 2, then 3, and so on. These numbers do not include zero or any negative numbers. So, they are a set of positive integers starting from 1.
For example, \[(1, 2, 3, 4, 5, ... )\].
Let's look at the set \(0, 3, 25\). Here, 3 and 25 are counting numbers because they are positive integers. However, 0 is not a counting number since we don't start counting from zero.
In summary, counting numbers are quite straightforward:

• They start from 1
• They include all positive integers after 1
• They do not include zero or negative numbers

Whole Numbers

Whole numbers expand on counting numbers by including one more important number: zero. They start from 0 and include all the positive integers as well. Whole numbers are practical because they help represent the absence of something (using 0).
For example, \[(0, 1, 2, 3, 4, 5, ... )\].
If we look at the set \(0, 3, 25\), all three numbers are whole numbers because they include zero and positive integers.
Therefore, whole numbers are very useful in various aspects of daily life and mathematics:

• They start from 0
• They include all positive integers
• They do not include any negative numbers or fractions

Non-Negative Integers

Non-negative integers are very similar to whole numbers. In fact, they are the same! Non-negative means that the numbers are either positive or zero, and integers mean they are not fractions or decimals.
Therefore, the set of non-negative integers includes all whole numbers. Taking the set \(0, 3, 25\), we see that all three numbers are non-negative integers. They are either zero or positive and, importantly, they are not fractions or decimals.
Summarizing non-negative integers:

• They include zero
• They include all positive integers
• They do not include negative numbers or fractions