Problem 36 - Algebra 1: Concepts and Skills

Question

Compare using \(<,>\) or \(=\) $$ 2.6 ? 2.65 $$

Step-by-Step Solution

Answer

2.6 < 2.65

Step 1: Comparing the decimal places

In 2.6 and 2.65, the part before the decimal point (the whole number part) is the same, which is 2. So, the comparison will depend on the numbers after the decimal point. In the number 2.6, it only has one decimal place which is 6 or \('6.0'\) if we add a '0' to compare with 2.65 that has two decimal places. Now, comparing 60 and 65, it is clear that 65 > 60.

Step 2: Final Comparison

Therefore, based on the conclusion from the previous step, 2.65 is greater than 2.6. So, we use the 'greater than' sign (>).

Key Concepts

Understanding Decimal Places

Decimal places represent the numbers found after the decimal point. They are crucial in determining the precise value of a number. In our example comparing the decimal numbers 2.6 and 2.65, you might notice that 2.6 has one digit after the decimal point, which is 6, while 2.65 has two digits, 6 and 5.

The number of decimal places helps us understand how many places we have for precision beyond the decimal point. When directly comparing decimal numbers, it’s helpful to make them have the same number of decimal places by adding zeros where necessary.

For example:

• 2.6 can be expressed as 2.60 to match the decimal places of 2.65.
• So instead of comparing 2.6 to 2.65, you are essentially comparing 2.60 to 2.65.

This method of matching decimal places simplifies the process, allowing for an accurate comparison. Once aligned, comparing them becomes straightforward!

Using Inequality Symbols

Inequality symbols are used in mathematics to show the relationship between two values. The most common inequality symbols are:

• "<" which means "less than"
• ">" which means "greater than"
• "=" which stands for "equals"

In our exercise with the numbers 2.6 and 2.65, after aligning the decimal places, we need to choose the correct inequality symbol.

Since 2.60 (or 2.6) is less than 2.65, we use the "less than" symbol: 2.60 < 2.65, which can also be written as 2.6 < 2.65.

Understanding which symbol to use is key in conveying the correct mathematical relationship. Inequality symbols streamline the comparison process, providing a clear visual representation of which value is larger or smaller.

Mathematical Comparison of Decimals

Mathematical comparison of decimals involves understanding the relative magnitude of decimal numbers. This process requires examining each digit, especially after the decimal point.

To compare 2.6 and 2.65, start by looking at the digits in descending order of significance. Both numbers have the same whole number "2". Next, take the first decimal place: both have the digit "6", so we continue to the next. Now, compare the second decimal place; here, 2.6 lacks an additional decimal while 2.65 has "5". Let's make 2.6 look like 2.60 to have a fair comparison.

These steps reveal that 2.60 does not reach the 2.65 value, thus proving 2.6 is smaller. Understanding how to systematically compare decimal numbers by examining decimal places helps solidify correct mathematical relationships and ensures accuracy in calculations.