Problem 84 - Prealgebra

Question

Determine which of the two given statements is true. ?0.081284 < ?0.08118 or ?0.081284 > ?0.08118

Step-by-Step Solution

Answer

?0.081284 > ?0.08118 is true.

Step 1: Align the Numbers

Align the two numbers ?0.081284 and ?0.08118 by their decimal points to ensure we compare each digit accurately. This means we write them as follows: Number 1: ?0.081284 Number 2: ?0.081180

Step 2: Compare Each Digit After the Decimal

Start comparing the digits in each position: - First digit after the decimal point is '0' for both numbers. - Second digit is '8' for both numbers. - Third digit is '1' for both numbers. - Fourth digit for Number 1 is '2', and for Number 2 it is '1'. Since 2 is greater than 1, ?0.081284 > ?0.08118.

Key Concepts

Place Value

Place value is a fundamental concept to understand when dealing with numbers, especially decimals. Every digit in a number has its own value depending on where it is located. This is true for whole numbers and decimal numbers alike.
When reading a decimal number, the digits to the left of the decimal point are whole numbers, and each holds a place value of ones, tens, hundreds, and so on. On the right side of the decimal point, the place values are parts of a whole and decrease by powers of ten:

• The first place is tenths (0.1).
• The second is hundredths (0.01).
• The third is thousandths (0.001).

Understanding place value is crucial when aligning numbers by their decimal points for comparison. It ensures that each digit is analyzed correctly based on its importance in the number. By recognizing where each digit sits, we can more accurately determine which of two decimal numbers is larger or smaller. For example, in the numbers ?0.081284 and ?0.08118, we correctly recognize the thousandths place by identifying that ?0.081284 extends one more place to ?0.081180, making alignment key.

Decimal Point

The decimal point is a small but powerful part of numerical notation. It distinguishes between whole numbers and fractional parts of a number. When you see a decimal point in a number, it separates the whole number on the left from the fraction represented by the digits on the right.
The importance of the decimal point cannot be overstated when comparing numbers. Everything to its right decreases in value exponentially, moving from tenths to thousandths as you move further right. This hierarchical importance makes the correct placement of the decimal point essential for precise digital representation and comparison.
Consider the two numbers in the exercise: ?0.081284 and ?0.08118. Aligning them correctly by the decimal point ensures that each respective place value is compared accurately. Without a consistent alignment, any comparison might yield incorrect conclusions. Hence, understanding the significance of the decimal point is vital for properly comparing decimals and not mistaking larger parts for smaller ones unexpectedly.

Digit Comparison

Digit comparison is an essential part of determining which of two decimal numbers is greater. This involves examining each corresponding digit from left to right after the decimal point, and understanding their value based on their position.
To effectively compare two decimals:

• Make sure both numbers are properly aligned at the decimal point.
• Start with the first digit after the decimal and compare corresponding digits in the same decimal place value.
• Continue this process until a difference is found.

In our example, ?0.081284 and ?0.08118, each digit was compared positionally:

• "0" in the tenths place for both.
• "8" in the hundredths place for both.
• "1" in the thousandths place for both.
• But at the fourth position, ?0.081284 has "2", while ?0.08118 has "1".

Because 2 is greater than 1, we know ?0.081284 is the larger number. By comparing digits in this ordered manner, you can consistently identify which decimal number is larger or smaller.