Question

Round off the following to three significant digits: (a) 20.155 (b) 0.204500 (c) 2055 (d) 0.2065

Short answer

(a) 20.2 (b) 0.205 (c) 2060 (d) 0.207

Step-by-step solution

Understand Significant Digits

Significant digits are the digits in a number that contribute to its precision. This includes all non-zero digits, zeros between non-zero digits, and trailing zeros in a decimal number.

Analyze Each Number

We'll look at each number one by one and identify the useful digits to decide how to round them correctly.

Round 20.155 to 3 Significant Digits

The number 20.155 has 5 significant digits. To round to 3 significant digits, look only at the first three non-zero digits, which are 20.1. The next digit, 5, indicates rounding up. Therefore, we round up the third digit: 20.2.

Round 0.204500 to 3 Significant Digits

Initially, identify the first three non-zero digits, which are 204. The number is currently 0.2045... To round, look at the fourth digit (5). Since it's 5 or more, round the last of the three significant digits up, so we get 0.205.

Round 2055 to 3 Significant Digits

Focus on the first 3 digits: 205. The next digit is 5, which means we should round up. So, 205 goes to 206. Therefore, 2055 rounded to three significant digits is 2060.

Round 0.2065 to 3 Significant Digits

Start from the left, identifying non-zero digits. First significant digits are 206. The next digit (5) is 5, so you must round the last significant digit up. This makes it 0.207.

Key concepts

Rounding Numbers

Rounding numbers is a technique used to simplify numbers by reducing them to a desired level of precision. This is often necessary to make numbers easier to understand and work with, especially when the exact value is not essential or when simplifying complex calculations.
When rounding, always check the digit right after the desired decimal or place you're rounding to. If this digit is 5 or greater, increase the last retained significant digit by one. If it's less than 5, leave the last retained digit unchanged. This ensures the number is as accurate as possible to the original.
For instance, rounding 20.155 to three significant digits involves focusing on the number 20.1 and the following digit 5, which tells us to round up to 20.2.

Precision in Measurements

Precision in measurements refers to how detailed and exact a measurement is. It's a crucial concept in fields like science, engineering, and statistics. A higher precision means that smaller changes in a measurement are detected, allowing for more detailed observations.
When we talk about significant digits in relation to precision, we're talking about the numbers that carry meaningful information about the measurement's accuracy. Each significant digit helps to verify the precision, telling us how reliable and exact our measurement is.
For instance, in the number 0.204500, there are four significant digits when rounded to 0.205. This maintains a high level of precision, ensuring that our calculations and measurements remain accurate within the constraints given.

Non-Zero Digits

Non-zero digits are any numbers that are not zero, such as 1, 2, 3, 4, 5, 6, 7, 8, and 9. These are always considered significant since they contribute to the value of a number and indicate the precision of the measurement.
In the process of rounding, they help determine how we simplify the number while maintaining its meaning. For example, when analyzing 2055 to find significant digits, we focus on non-zero values to guide the rounding process, resulting in the precise number of 2060 when rounded to three significant digits.
Understanding how non-zero digits function within a number helps in ensuring accurate and meaningful simplifications during rounding, reflecting the true nature of the measurement.