Problem 28 - Technical Mathematics with Calculus

Question

Round each number to two decimal places. $$55.8650$$

Step-by-Step Solution

Answer

55.8650 rounded to two decimal places is 55.87.

Step 1: Identify the Decimal Place

Determine which digit is in the hundredths place. For the number 55.8650, the digit in the hundredths place is 6.

Step 2: Locate the Next Digit

Look at the digit to the right of the hundredths place to decide whether to round up or stay the same. The digit to the right of 6 is 5.

Step 3: Apply Rounding Rules

Since the digit to the right (5) is 5 or greater, round the hundredths place up by one. The 6 becomes 7.

Step 4: Finalize the Rounded Number

After rounding, the number to two decimal places is 55.87.

Key Concepts

Decimal Places

Understanding decimal places is fundamental when working with numbers that have a fractional component. A decimal point separates the integer part of a number from its fractional part. Each position after the decimal point represents a 'place', counting tenths, hundredths, thousandths, and so on. For example, in the number 0.75, the 7 is in the tenths place, and the 5 is in the hundredths place.

When you're asked to round a number to a specific number of decimal places, like two decimal places, you're being instructed to adjust the number so that it shows only two digits after the decimal point. Acknowledging the value of each decimal place is important because it determines the precision of the number; the more decimal places, the more precise the number. Precision is essential in fields such as science, finance, and engineering, where accurate measurements are crucial.

Rounding Rules

Rounding rules help us convert a number into a more manageable form while maintaining its value as close to the original as possible. The basic principle is to increase the value of the last retained digit if the next digit (the one immediately after) is 5 or greater, commonly referred to as rounding up. If this next digit is less than 5, the last retained digit stays the same - this is known as rounding down.

Let's examine the rounded to two decimal places example, \(55.8650\). The third digit after the decimal is 5, which triggers the rounding up of the second digit (6). Hence, \(55.8650\) rounded to two decimal places is \(55.87\). Always remember after rounding, any digits beyond the specified decimal place are dropped. These rules ensure consistency in calculations and are critical when adjusting figures to a desired level of precision.

Significant Figures

Significant figures, or 'sig figs' as they are sometimes known, refer to the digits in a number that carry meaning contributing to its measurement precision. This concept differs from decimal places because significant figures consider all the numbers, not just those after the decimal point. In the number \(55.8650\), for example, all digits are significant except the last zero; it's merely a placeholder and does not contribute to precision.

When rounding to significant figures, the rules are similar - look at the digit immediately after the last significant figure you want to retain. If this digit is 5 or more, round up. If less, do nothing. Significance plays a vital role in sciences, where measuring and recording data with the correct precision is necessary for accuracy and reliability. Important to remember is that zeros can sometimes be significant especially if they come after non-zero digits and after the decimal point.