Problem 3 - Beginning and Intermediate Algebra

Question

Insert \(<,>,\) or \(=\) in the appropriate space to make the statement true. See Example 1. $$ 6.26 \quad 6.26 $$

Step-by-Step Solution

Answer

Both numbers are equal; insert '='.

Step 1: Identify the Problem

We need to determine if the two numbers are equal, or if one is greater than the other by comparing 6.26 and 6.26.

Step 2: Compare Decimal Places

Look at the whole numbers first. Here, both numbers have '6' before the decimal point, which are equal.

Step 3: Compare Tenths Place

Both numbers have the digit '2' in the tenths place, making them equal as well.

Step 4: Compare Hundredths Place

For the hundredths place, both numbers have the digit '6'. These are also equal.

Step 5: Conclusion on Comparison

Since both numbers have the same digits in all corresponding places, they are equal.

Step 6: Insert Appropriate Symbol

We conclude that 6.26 is equal to 6.26, so we insert '=' between them.

Key Concepts

Comparing Decimals

Understanding how to compare decimals is a crucial skill in mathematics. This involves looking at numbers that include decimal points and determining if one is greater than, less than, or equal to another. To start, compare the numbers beginning from the leftmost digit at the same place value.

• Look at the whole number part of the decimal. If one is greater, then that number is larger.
• If the whole numbers are equal, evaluate the digits to the right of the decimal point.

This side-by-side comparison continues until a difference is found. If every digit matches, the numbers are equal. For example, in comparing the numbers 6.26 and 6.26, you start by examining the whole numbers (both are 6), then each subsequent decimal place. Since every matched digit is identical, the two numbers are equal.

Decimal Places

Decimal places refer to the digits that appear after the decimal point in a number. These digits represent fractions of whole numbers. Each decimal place has its own value:

• The first digit after the decimal is the tenths place, equivalent to \(\frac{1}{10}\).
• The second is the hundredths place, represented by \(\frac{1}{100}\).
• The third is the thousandths place, which is \(\frac{1}{1000}\), and so on.

The importance of decimal places is evident when it comes to precision. Even a small change, such as from 6.26 to 6.27, can make a significant difference. When comparing decimals, check each place thoroughly, starting from the tenths and proceeding to finer decimal points as needed.

Equality in Decimals

For two decimals to be considered equal, every digit from the whole number to the farthest decimal place checked must match exactly. This means that there should be no difference in any of the corresponding digits at any decimal place checked. When considering equality, the overall value remains unchanged regardless of the number of zeros placed after the last non-zero digit. For instance, 6.260 and 6.26 are equal because the extra zero does not affect the value of the number, as it adds no additional value. Understanding equality in decimals is crucial since it ensures exactness and eliminates ambiguity when performing calculations and making decisions based on those calculations.