Problem 25 - Prealgebra

Question

Place either < or \(>\) between each of the following pairs of numbers so that the resulting statement is true. $$-0.75 \quad 0.25$$

Step-by-Step Solution

Answer

-0.75 < 0.25

Step 1: Understanding the Numbers

First, let's understand what the numbers -0.75 and 0.25 are. -0.75 is a negative number, while 0.25 is a positive number. In number comparisons, any negative number is always less than a positive number.

Step 2: Decision on Inequality Symbol

Since -0.75 is a negative number and 0.25 is a positive number, it follows that -0.75 is smaller than 0.25. Therefore, we need to use the 'less than' symbol: <.

Step 3: Writing the Correct Statement

Place the 'less than' symbol between the numbers to form the statement: \[ -0.75 < 0.25 \]. This statement correctly reflects the order of these numbers on the number line.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero. Imagine a number line that extends infinitely in both directions, with zero in the middle. Negative numbers are found to the left of zero.

• The farther left a number is, the smaller it is.
• Negative numbers are represented with a minus sign (-) in front of them.

-0.75 is an example of a negative number. It's less than zero and therefore less than any positive number.
Unlike positive numbers, which count upwards, negative numbers measure what you owe or lack. So, in any comparison between negative and positive numbers, negatives are always smaller.

Exploring Positive Numbers

Positive numbers are parts of basic arithmetic that are vital to mathematical understanding. These are numbers greater than zero and appear to the right of zero on the number line.

• They're usually marked with a plus sign (+) or no sign at all.
• Examples include fractions such as 0.25, and whole numbers like 1, 2, or 3.

Positive numbers signify a gain, growth, or addition. In comparing numbers, a positive number will always be greater than a negative number.
This is why, when you have to compare -0.75 (a negative) and 0.25 (a positive), 0.25 will always be larger.

Number Line Comparison

The number line is a visual representation that helps us understand the order and size of numbers. It is essentially a straight line with numbers placed at equal intervals along its length.

• Zero acts as the central point on this line, dividing positive numbers on the right and negative numbers on the left.
• To compare numbers, locate them on the number line to see which is further right or left.

Using the Number Line to Compare

When comparing -0.75 to 0.25, place both numbers on a number line.
You'll notice -0.75 is on the left, and 0.25 is on the right. This placement shows that -0.75 is indeed less than 0.25, supporting the use of the "less than" symbol: \( -0.75 < 0.25 \).
Number lines help visually cement the idea that negatives are smaller than positives and assist in making correct comparisons in prealgebra.