Problem 25 - Fundamentals of Mathematics

Question

For the pairs of real numbers in the following 5 problems, write the appropriate symbol \((<,>,=)\) in place of the \(\square\) $$ 10 \square 10 $$

Step-by-Step Solution

Answer

10 = 10

Step 1: Understand the Question

We need to compare the two numbers given, which are both 10, to determine their relationship using symbols '<', '>', or '='.

Step 2: Identify the Relationship

Since the two numbers we are comparing are the same (10 and 10), we need to determine if one number is greater than, less than, or equal to the other.

Step 3: Apply the Equality Symbol

Given that both numbers are identical, we apply the '=' symbol to indicate that 10 is equal to 10, i.e., 10 = 10.

Key Concepts

Equality Symbol

In mathematics, the equality symbol "=" is used to show that two values are the same. This is a crucial concept when comparing real numbers, as it communicates equivalence between them. Consider two numbers like in the original exercise: 10 and 10. By observing these numbers, we can see they are identical. Therefore, we use the equality symbol.

When you use "=", you are saying:

• The two quantities are exactly equal.
• There is no difference between them in value.

Remember, the equality symbol is used widely, not just in numbers but also in equations, expressions, and more complex mathematical notions. It sets the foundation in math for expressing when things are the same.

Inequality Symbols

While the equality symbol represents equivalence, inequality symbols are used to show how numbers differ. In mathematics, we commonly use two inequality symbols: "<" for less than, and ">" for greater than. These symbols help us to characterize the relationship between numbers that are not equal.

When we compare real numbers, we ask:

• Is one number larger than the other?
• Is one number smaller?

For instance, if you compare 5 and 10, you would write 5 < 10, meaning 5 is less than 10. Conversely, with 15 and 10, you would write 15 > 10 to indicate that 15 is greater than 10. Inequality symbols help compare and order numbers systematically.

Real Number Comparison

Comparing real numbers is fundamental in solving problems across mathematics. Real numbers include all the numbers you can think of: whole numbers, fractions, and irrational numbers like \( \sqrt{2} \) and \( \pi \). During real number comparison, we determine whether two numbers are equal, or one is greater or lesser than the other.

Here’s how:

• If two numbers are exactly the same, use "=".
• If one number is bigger, use ">".
• If one number is smaller, use "<".

By making these comparisons, you can solve equations, inequalities and analyze data sets. This lays the groundwork for further advanced mathematical concepts and applications.