Round each decimal to the nearest thousandth. (a) 0.8 (line above 8) (b) 0.4 (line above 4) (c) 0.9762 (d) 0.8645

Verified step by step guidance

1. Understand that rounding to the nearest thousandth means keeping three digits after the decimal point and deciding whether to round the last digit up or keep it the same based on the digit immediately after it.
2. For each decimal, identify the digit in the thousandths place (the third digit after the decimal point). If the number has fewer than three digits after the decimal, consider adding zeros to reach the thousandths place.
3. Look at the digit immediately to the right of the thousandths place (the ten-thousandths place). If this digit is 5 or greater, increase the thousandths digit by 1; if it is less than 5, leave the thousandths digit as is.
4. Apply this rule to each decimal: (a) 0.8 (with a repeating 8), (b) 0.4 (with a repeating 4), (c) 0.9762, and (d) 0.8645, carefully identifying the thousandths and ten-thousandths digits for each.
5. Write the rounded number for each part, ensuring it has exactly three digits after the decimal point, reflecting the rounding decision made in the previous step.

Key Concepts

Place Value and Decimal Notation

Understanding place value is essential for rounding decimals. Each digit in a decimal number has a specific place value, such as tenths, hundredths, and thousandths. Recognizing these positions helps determine which digit to round to and which digit influences the rounding.

Rounding Rules for Decimals

Rounding decimals involves looking at the digit immediately to the right of the target place value. If this digit is 5 or greater, increase the target digit by one; if less than 5, keep the target digit the same. This process simplifies numbers while maintaining approximate value.

Repeating Decimals and Their Representation

Repeating decimals have one or more digits that repeat infinitely, often indicated by a line over the repeating digit(s). When rounding repeating decimals, it is important to consider the repeating pattern to accurately approximate the value to the desired decimal place.