Question

Round each of the following numbers to four significant figures, and express the result in standard exponential notation: (a) \(102.53070\), (b) 656,980 , (c) \(0.008543210\), (d) \(0.000257870,(\mathrm{e})-0.0357202 .\)

Short answer

The numbers rounded to four significant figures and expressed in standard exponential notation are: (a) \(1.025 \times 10^2\) (b) \(6.569 \times 10^5\) (c) \(8.543 \times 10^{-3}\) (d) \(2.579 \times 10^{-4}\) (e) \(-3.572 \times 10^{-2}\)

Step-by-step solution

Identifying Significant Figures

The significant figures in a number are the digits that are known with certainty plus one uncertain digit. In this exercise, we want to round each of the given numbers to four significant figures. (a) 102.53070 (b) 656,980 (c) 0.008543210 (d) 0.000257870 (e) -0.0357202 ##Step 2: Rounding to Four Significant Figures##

Rounding to Four Significant Figures

We'll round each number so that they have four significant figures. Remember that any zeros before the first nonzero digit are not considered significant. (a) 102.5 (b) 656,900 (c) 0.008543 (d) 0.0002579 (e) -0.03572 ##Step 3: Converting to Standard Exponential Notation##

Converting to Standard Exponential Notation

Now we'll convert each rounded number into standard exponential notation, which shows the number as a product of two numbers: a coefficient between 1 and 10, and 10 raised to some exponent. (a) \(1.025 \times 10^2\) (b) \(6.569 \times 10^5\) (c) \(8.543 \times 10^{-3}\) (d) \(2.579 \times 10^{-4}\) (e) \(-3.572 \times 10^{-2}\) The numbers rounded to four significant figures and expressed in standard exponential notation are: (a) \(1.025 \times 10^2\) (b) \(6.569 \times 10^5\) (c) \(8.543 \times 10^{-3}\) (d) \(2.579 \times 10^{-4}\) (e) \(-3.572 \times 10^{-2}\)

Key concepts

Rounding Numbers

Rounding numbers is an essential skill in mathematics and science, particularly when dealing with measurements or data values. When you round a number, you alter it slightly to make it simpler and easier to use. The goal is to keep the value of the number close to what it originally was, while having fewer digits. When rounding to a specific number of significant figures, you start counting from the first non-zero digit. You round up if the digit after your last significant figure is 5 or more, and round down if it is less. Let's look at an example: to round 102.53070 to four significant figures, we consider the first four digits, which are 102.5. We look at the 3 in the hundredths place — since it is smaller than 5, we do not round up. So, the rounded number is 102.5. Remember, zeros may or may not be significant based on their position. Leading zeros (those before the first non-zero digit) are not significant, while trailing zeros (those after the decimal but before a non-zero digit) can be significant, according to context. This distinction is crucial for accurate rounding.

Scientific Notation

Scientific notation is a way of expressing extremely large or small numbers conveniently. It allows scientists to handle numbers without writing out all the zeros, making calculations easier and less error-prone. In scientific notation, a number is expressed as a value between 1 and 10 multiplied by a power of 10. For example, the number 656,900 becomes \(6.569 \times 10^5\). The key here is to move the decimal point in the original number to create a new number, known as the 'coefficient', which is between 1 and 10. You then count how many places you moved the decimal point to determine the exponent on the 10. If you move the decimal to the left, the exponent is positive; if to the right, it is negative.

Exponential Notation

Exponential notation is similar to scientific notation, but often used when discussing powers of numbers more generally. It focuses on simplifying multiplication processes, especially when numbers are raised to powers. In our exercise, we used exponential notation to express numbers like \(1.025 \times 10^2\). For 102.5, the scientific notation gives us a clean, simple presentation, which is \(10^2\), emphasizing how often 10 is multiplied by itself. The idea of an exponent is crucial not only for simplifying multiplication but also for understanding relationships in equations, growth/decay problems, and more. Selecting the correct exponent involves understanding the base (which in our case is 10) and how many times it is used to reach the original number.

Precision In Measurements

Precision in measurements refers to how exact a measurement is. The more significant figures, the more precise the measurement. However, precision is not the same as accuracy, which refers to how close a measurement is to the true value. Using significant figures correctly assures that calculations reflect the precision of the measured values. For example, expressing 0.008543210 to four significant figures results in 0.008543. This maintains precision without implying an unwarranted level of exactness beyond what's measured. Choosing the right level of precision and understanding significant figures in calculations help avoid misleading representations of data. It informs how scientists report and interpret experimental results, ensuring they communicate the correct degree of confidence in their measurements.