Problem 89 - Prealgebra

Question

Determine which of the two given statements is true. 36.8298 < 36.8266595 or 36.8298 > 36.8266595

Step-by-Step Solution

Answer

36.8298 > 36.8266595 is true.

Step 1: Identify the Given Numbers

The given numbers are 36.8298 and 36.8266595. We need to compare these two numbers.

Step 2: Compare the Decimal Parts

Both numbers have the same integer part, which is 36. Therefore, compare the decimal parts: 0.8298 and 0.8266595.

Step 3: Compare Decimal Parts Digit by Digit

Start comparing from the first digit after the decimal point: - First decimal digit: 8 in 0.8298 is greater than 8 in 0.8266595, so we continue. - Second decimal digit: 2 in 0.8298 is greater than 2 in 0.8266595, so we continue. - Third decimal digit: 9 in 0.8298 is greater than 6 in 0.8266595.

Step 4: Conclusion of Comparison

Since 0.8298 has a greater third decimal digit (9) than 0.8266595 (6), the number 36.8298 is greater than 36.8266595.

Key Concepts

Understanding Decimal Numbers

Decimal numbers are a way to represent fractions of a whole using the base-10 system. A decimal number consists of a whole number part and a fractional part, separated by a decimal point.

• For example, in the number 36.8298, "36" is the whole number part and ".8298" is the fractional part.
• Decimals allow us to express numbers more precisely than whole numbers. This is crucial for comparing numbers that are very close in value.

Decimal numbers are especially useful in measurements and finance, where precision is important.
When working with decimals, it is critical to understand how each digit represents a different place value, which impacts how we compare and manipulate these numbers.

Grasping Place Value

The concept of place value is essential in understanding decimal numbers. Each digit in a decimal number has a place, and each place has a value based on its position relative to the decimal point.

• The digits to the left of the decimal point represent whole numbers, increasing by powers of ten as you move left. For instance, in the number 36.8298, the "3" is in the tens place, meaning 3 tens or 30, and the "6" is in the units place, meaning 6 ones or 6.
• The digits to the right of the decimal point represent fractional values, decreasing by powers of ten as you move right. For example, the first decimal place is tenths, so the "8" in the tenths place (in 0.8298) is worth 0.8.

Understanding place value helps us compare decimal numbers more effectively, as it allows us to evaluate each digit's significance in relation to others.

Recognizing Inequalities

Inequalities are mathematical statements indicating that two values are not equal, and they help us compare numbers to see which is greater or smaller.

• Symbols used in inequalities include "<" (less than) and ">" (greater than).
• In our exercise, we compared 36.8298 and 36.8266595 using these symbols.

We started by comparing the integer parts, which were the same, and then moved to the decimal parts.
This process of comparing digit by digit ensures accuracy.

• If one digit in the decimal place is higher in value than the corresponding digit in another number, the entire number can be deemed greater.
• This method elucidates why 36.8298 is greater than 36.8266595; the digit "9" in the thousandths place is larger than "6" in the same place of the other number.

Understanding these steps is vital in unlocking the power of inequalities for comparing decimal numbers in a detailed and meaningful way.