Problem 71 - Algebra 1

Question

Compare using \(<,=,\) or \(>.\) $$0.333 \underline{?} 0.3333$$

Step-by-Step Solution

Answer

0.333 < 0.3333

Step 1: Understand the number system

When comparing decimal numbers, it is important to understand that the value of a digit becomes less as it moves farther from the decimal point to the right. The first place to the right of the decimal point is the tenths place, the second is the hundredths place, the third is the thousandths place, and so on.

Step 2: Make the decimal lengths uniform

The decimal lengths should be made uniform in order to proceed with the comparison. This can be done by appending zero(s) to the less precise number. Here, the number 0.333 can be written as 0.3330.

Step 3: Compare the numbers

Now compare the numbers by observing each decimal place from left to right. When we compare 0.3330 with 0.3333, it is clear that 0.3330 has 0 at the fourth decimal place, while 0.3333 has 3 at the same place. Hence, 0.3330 is less than 0.3333.

Key Concepts

Understanding Decimal Places

Decimal places refer to the digits located to the right of the decimal point in a number. Each position to the right has a specific place value that becomes smaller as you move further away from the decimal point.
For example:

• The first position is the tenths place.
• The second position is the hundredths place.
• The third position is the thousandths place.

These decimal places allow us to express values smaller than one whole unit.
When writing or reading decimal numbers, it's essential to recognize the importance of each place value, as it influences the overall value of the number.
This understanding is crucial when performing operations such as addition, subtraction, or comparison.

Effective Number Comparison

Comparing decimal numbers involves examining each digit's place value starting from the left, just after the decimal point.
This sequential comparison helps determine if one number is larger, smaller, or equal to another.
Here's how you can compare two decimal numbers effectively:

• Align the numbers vertically by the decimal point.
• Compare the digits in each place value, starting from the left.
• If one number has more digits, add zeros to the shorter one without changing its value.

Once this is done, proceed to compare the numbers place by place.
If at any point one digit is greater than the corresponding digit in the other number, it shows which number is larger.
For instance, 0.333 is shorter than 0.3333. Writing 0.333 as 0.3330 helps easily identify its smaller size, especially when comparing them.

Decimal Place Value Importance

Understanding place values in decimals is paramount to accurately working with decimal numbers.
In decimals, each digit represents a fraction of a whole number, with its specific place value indicating how much of the fraction it represents.
A helpful way to think of place value in decimals is as a fraction of 10:

• The tenths place (\(0.1\) fraction)
• The hundredths place (\(0.01\) fraction)
• The thousandths place (\(0.001\) fraction)

Working with decimals requires precise understanding because even a small change in a single decimal place can significantly alter the number's value.
In comparing or converting decimals, knowing the correct place value helps maintain consistency and accuracy, ensuring that calculations reflect true values.