Problem 84 - College Algebra

Question

If you are given two real numbers, explain how to determine which one is the lesser.

Step-by-Step Solution

Answer

To identify the lesser of two real numbers, represent them on the number line. If both numbers are positive, the one further to the left is lesser. For negative numbers, the one further to the left is also lesser. Zero is less than positive numbers and more than negative numbers. Lastly, negative numbers are always lesser than positive numbers.

Step 1: Understand the number line

Real numbers can be represented on the number line. On this line, the values increase towards the right and decrease towards the left. The numbers to the left of zero are negative and those to the right are positive.

Step 2: Compare positive numbers

If both numbers are positive and one number is to the right of the other on the number line, then that number is greater. The other number is lesser.

Step 3: Compare negative numbers

If both numbers are negative, the one further to the left on the number line is lesser.

Step 4: Zero comparison

Any positive number is greater than zero and any negative number is less than zero.

Step 5: Negative and positive comparison

If one number is negative and the other is positive, the negative number is always lesser.

Key Concepts

Number Line

The number line is a simple yet powerful tool in mathematics that helps visualize real numbers. Picture it as a straight horizontal line with equally spaced markings. This line has a middle point usually marked as 0. As you look from left to right on the number line, the values increase from negative to positive.
It's like a visual map where:

• All numbers on the left side of zero are negative numbers.
• All numbers on the right side of zero are positive numbers.

This simple framework provides a clear way to see and compare different numbers, making it easier to identify which numbers are larger or smaller just by their positions.

Positive and Negative Numbers

Positive and negative numbers form the backbone of the real number system. On the number line, positive numbers are always situated to the right of zero. Conversely, negative numbers reside to the left.
Understanding these positions helps immensely in organizing numbers:

• Positive numbers represent values greater than zero.
• Negative numbers indicate values less than zero.

For example, +3 is a positive number and appears to the right of zero, while -3, a negative number, can be found on the left. Recognizing the position of these numbers aids in performing operations or comparisons with ease.

Comparison of Real Numbers

When comparing real numbers, their position on the number line provides valuable insight. Every number has a unique position, which simplifies determining which is greater or lesser. Here are some key points:

• For two positive numbers, the one further to the right is greater.
• For two negative numbers, the one further to the left is lesser.
• Any positive number is always greater than zero, while any negative number is lesser than zero.
• If you compare a negative number with a positive number, the negative is always lesser.

This basic understanding of number positions helps us confidently decide the relative size of any two real numbers whether positive or negative.