Problem 9 - Fundamentals of Mathematics

Question

Write the appropriate symbol \((<,>,=)\) in place of the \(\square\). $$ -7 \square 0 $$

Step-by-Step Solution

Answer

-7 < 0

Step 1: Understand the Problem

We need to compare two numbers: -7 and 0, by filling in the appropriate comparison symbol (<, >, or =) in the blank.

Step 2: Know the Integer Line

Recall that on the integer line, numbers increase as you move to the right. Negative numbers are to the left of zero.

Step 3: Compare the Numbers

Since -7 is to the left of 0 on the number line, it means -7 is less than 0.

Step 4: Write the Correct Symbol

Based on our comparison, place the symbol "<" in the square: \(-7 < 0\).

Key Concepts

Number Line

A number line is a visual tool that helps us understand the order and position of numbers. It is a horizontal line where numbers are placed at equal intervals. Imagine it like a ruler that extends infinitely in both directions. On a number line, numbers increase as you move right and decrease as you shift left.

Here are some key points about the number line:

• Positive numbers are located to the right of zero.
• Negative numbers are on the left of zero, showing they are less than zero.
• Zero acts as a central point or a boundary between positive and negative numbers.

Using a number line, we can easily compare integers by seeing which number is further left or right. In essence, if one number is to the left of another, it is smaller; if it is to the right, it is larger.

Negative Numbers

Negative numbers are those that are less than zero. These numbers appear to the left of zero on the number line. They represent values that are opposite to their positive counterparts. For example, in the context of temperature, negative values would mean subzero or degrees below freezing.

Some important aspects of negative numbers are:

• They indicate a lack or deficit in many real-world contexts, like debt.
• The further a negative number from zero, the smaller its value. So, -10 is smaller than -3.
• Negative numbers can also indicate directional movements or positions when talking about coordinates.

Understanding negative numbers is crucial for properly placing and interpreting integers on a number line.

Comparison Symbols

Comparison symbols are used to evaluate and describe the relationship between two numbers. The primary symbols used are:

• ">" (greater than): This symbol is used when the number on the left is larger than the number on the right.
• "<" (less than): This symbol indicates that the number on the left is smaller than the number on the right.
• "=" (equal to): This is used when both numbers are the same.

To correctly compare integers, position them on the number line. If one number is to the left of another, it is less, thus you use "<". Conversely, if it is to the right, you apply ">". And, if both are at the same point, the "=" sign reflects equality.

When comparing -7 and 0, as shown on the number line, -7 is leftward of 0, meaning -7 is less. Therefore, the correct symbol is "<".