Question

Correct any answers that have the incorrect number of significant figures. (a) \((78.56-9.44) \times 45.6=3152\) (b) \(\left(8.9 \times 10^{5} \div 2.348 \times 10^{2}\right)+121=3.9 \times 10^{3}\) (c) \((45.8 \div 3.2)-12.3=2\) (d) \(\left(4.5 \times 10^{3}-1.53 \times 10^{3}\right) \div 34.5=86\)

Short answer

(a) 3160, (b) 3.79 \times 10^{3}, (c) 2.01, (d) 86.1

Step-by-step solution

- Correcting (a)

Calculate the difference and then multiply: (78.56-9.44) = 69.12. Now multiply by 45.6: 69.12 × 45.6 = 3152.912, which we round to 3160 because the minimum number of significant figures in any of the numbers in the calculation is 3.

- Correcting (b)

Perform division, then multiplication, and finally addition: (8.9 × 10^5) ÷ (2.348 × 10^2) = 3.789 × 10^3. Add 121, which is considerably less significant and does not affect the figure with the highest exponent: 3.789 × 10^3 + 121 ≈ 3.789 × 10^3, which should be rounded to 3.79 × 10^3 for three significant figures.

- Correcting (c)

Divide, then subtract: (45.8 ÷ 3.2) = 14.3125. Then subtract 12.3: 14.3125 - 12.3 = 2.0125, which should be rounded to 2.01 to match the three significant figures in 12.3.

- Correcting (d)

Subtract, then divide: (4.5 × 10^3) - (1.53 × 10^3) = 2.97 × 10^3. Then divide by 34.5: 2.97 × 10^3 ÷ 34.5 ≈ 86.087, which should be rounded to 86.1 to match the three significant figures in 1.53 × 10^3.

Key concepts

Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It's like shorthand for numbers, allowing us to write them in a format that is both compact and precise. In scientific notation, a number is expressed as the product of two numbers: a coefficient and 10 raised to a power. The coefficient must be greater than or equal to 1 and less than 10, and it includes all the significant figures of the original number. For example, the number 8.9 × 10^5 represents a coefficient of 8.9, which has two significant figures, and indicates that the decimal point has been moved 5 places to the right.

Using scientific notation is especially useful in science and engineering to handle the extremely large or small quantities often encountered in calculations. Understanding this concept helps you maintain precision in your calculations without losing track of the magnitude of your numbers.

Arithmetic Operations with Significant Figures

When performing arithmetic operations, the rule of significant figures must be followed to ensure that the precision of the result is not overestimated. Each step of a calculation must consider the number of significant figures. For example, when multiplying or dividing, the answer must have the same number of significant figures as the number in the calculation with the fewest significant figures.

In addition or subtraction, the result should have the same number of decimal places as the number with the fewest decimal places. Understanding these rules is crucial because it helps preserve the integrity of the calculation's precision and prevents the introduction of implied accuracy that isn't supported by the original data.

Rounding Numbers

Rounding numbers is the process of simplifying a number while still keeping its value close to what it was. The purpose is to make the number easier to work with. Usually, rounding is done when the exact value is not as important as having a value that's easy to use. In the context of significant figures, rounding is often necessary after a calculation is performed. The key idea is to ensure that the final answer is not more precise than the least precise measurement used in the calculation.

Rounding follows specific rules, such as rounding up if the next digit is 5 or higher, and rounding down if it's less than 5. When rounding numbers in scientific notation, the same principles apply, but the focus is on the decimal part of the number, the coefficient.

Significant Figures Rules

Significant figures (often abbreviated as 'sig-figs') are the digits in a number that carry meaning contributing to its precision. This includes all the non-zero digits, any zeros between them, and any trailing zeros in a decimal number. The rules of significant figures serve as a guideline for determining the number of meaningful digits in a given number and dictate how to maintain precision during calculations.

For instance, when multiplying or dividing, the number of significant figures in the result should be the same as that in the measurement with the smallest number of significant figures. When adding or subtracting, the result should be rounded off to the least significant decimal place of any number in the operation. By respecting these rules, scientists and engineers can communicate their findings with the appropriate level of precision.