Question

Round off the following to three significant digits: (a) 10.25 (b) 10.20 (c) 0.01029 (d) 10,248

Short answer

(a) 10.3, (b) 10.2, (c) 0.0103, (d) 10,200

Step-by-step solution

Identify Significant Digits

Significant digits are the numbers in a value that contribute to its precision. For each number given, identify the first three significant digits for rounding.

Evaluate and Round 10.25

The significant digits in 10.25 are 1, 0, and 2. The next digit is 5, which is 5 or greater. Therefore, we round the last significant digit up from 2 to 3. The number becomes 10.3.

Evaluate and Round 10.20

In 10.20, the significant digits are 1, 0, and 2, because trailing zeros in the decimal part are significant. The next digit is 0. Since it is less than 5, we leave the digit 2 unchanged. Thus, it remains 10.2.

Evaluate and Round 0.01029

For 0.01029, the significant digits are 1, 0, and 2, starting from the first non-zero digit. The next digit is 9, which is 5 or greater, so we increase the last significant digit by 1. This makes the number 0.0103.

Evaluate and Round 10,248

In 10,248, the significant digits are 1, 0, and 2. The digit following is 4, which is less than 5, leaving the last significant digit 2 unchanged. Thus, it rounds to 10,200.

Key concepts

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify numbers to make them easier to work with. This involves reducing the number of significant figures in a number while maintaining its general size and magnitude, making calculations quicker and comparisons simpler.

The basic principle of rounding numbers is deciding which digit to keep and which to adjust. Here’s what you typically do:

• If the digit following your last significant figure is 5 or greater, you round up.
• If it is less than 5, you round down.

This process is essential in scientific and technical fields where overly precise numbers can become cumbersome. For instance, rounding 10.25 with significant digits 1, 0, and 2 gives you 10.3 since the next digit is 5, prompting an increase.

Rounding provides a neat way to simplify data without losing valuable information.

Decimal Places

Decimal places refer to the number of digits that appear after the decimal point in a number. This concept is crucial in determining the accuracy of calculations and measurements.

Identifying how many decimal places you need depends on the context:

• In finance, more decimal places might be used to account for cents and fractions of a cent.
• In science, the number of decimal places may indicate the precision of measurements.

For the number 10.20 in our exercise, identifying decimal places helps us establish that trailing zeros after a decimal point contribute to the number's precision. Thus, 10.20 is not just 10.2, as 10.20 communicates a specific precision level where the zero is significant.

Understanding decimal places helps prevent errors in scientific calculations and financial transactions.

Precision in Measurements

Precision in measurements reflects how exact and detailed a measurement is. It indicates how close repeated measurements of the same quantity are to one another.

Often marked by the number of significant digits or decimal places, precision is not the same as accuracy, which measures how close a measurement is to the true or accepted value.

For example, when rounding the value 0.01029, noting that 9, being 5 or greater, rounds the number to 0.0103. Here, three significant digits convey a specific precision level. The number of significant figures, therefore, signals the precision of the measurement.

It's crucial to understand that while precision reflects detail and consistency, maintaining an appropriate level of precision ensures efficiency without unnecessary complexity.