Problem 10 - Fundamentals of Mathematics

Question

Write the appropriate symbol \((<,>,=)\) in place of the \(\square\). $$ -1 \square 0 $$

Step-by-Step Solution

Answer

-1 < 0

Step 1: Understand the Numbers

The problem involves comparing two numbers: -1 and 0. We need to determine the relationship between these two numbers in terms of less than, greater than, or equal to.

Step 2: Analyze the Numeric Order

In any number line, negative numbers are always less than positive numbers and zero. Since -1 is a negative number and 0 is non-negative, we can directly use the rule that a negative number is less than zero.

Step 3: Place the Appropriate Symbol

Since -1 is less than 0, the correct symbol to use is '<'. Thus, replacing \(\square\), we write: -1 < 0.

Key Concepts

Number Line

Understanding a number line can greatly help when comparing numbers. Picture a horizontal line with numbers placed at equal intervals along it. The center of this line usually hosts the number zero. Positive numbers dwell to the right of zero, increasing as you move further right.
Conversely, negative numbers reside to the left of zero, decreasing as you move left. The number line serves as a visual representation of numeric order.

• Zero is the midpoint between negative and positive numbers.
• Numbers increase from left to right.
• Negative numbers are always positioned to the left of zero.

When comparing numbers using a number line, remember that the further right a number is, the greater it is.

Negative Numbers

Negative numbers can often be puzzling, but they are quite simple to grasp. They are numbers less than zero, represented with a minus sign (−) in front of them, such as -1, -2, or -3. On a number line, negative numbers lie to the left of zero, meaning they are always lesser than zero and any positive number. Here's why negative numbers are important to understand:

• They indicate quantities below zero, like debts or temperatures below freezing.
• Negative numbers are inherently smaller than any non-negative number.
• Understanding them is crucial for many real-world problems.

When you compare negative numbers, remember that a number further left is smaller—for instance, -3 is less than -1.

Less Than Symbol

The less than symbol, denoted as "<", is used to express that one number is smaller than another. When comparing two numbers, place the less than symbol between them to illustrate the first number is smaller than the second one. For example, -1 < 0 shows -1 is less than 0. A few points to remember about the less than symbol:

• It is used for comparing size or value of numbers.
• On a number line, the "<" symbol indicates the number on the left is smaller.
• It helps to easily visualize the relationship between numbers.

Using the less than symbol correctly involves understanding its orientation—the open end should always point towards the greater number.