Question

Express each of the following to fur significant figures. (a) 3984.6 (b) 422.04 (c) 0.0033 (d) 902.10 (e) 0.02173 (f) 7000 (g) 7.02 (h) 67,000,000

Short answer

The numbers expressed to four significant figures are: (a) 3985, (b) 422.0, (c) 0.0033, (d) 902.1, (e) 0.02173, (f) 7000, (g) 7.02, (h) 67,000,000.

Step-by-step solution

Identify the significant figures of 3984.6

All non-zero numbers are considered significant. So, 3984.6 has five significant figures. To express this as four significant figures, round it off. Using rounding rules, 3984.6 becomes 3985.

Identify the significant figures of 422.04

All non-zero numbers are considered significant. So, 422.04 also has five significant figures. To express this as four significant figures, round it off. Using rounding rules, 422.04 becomes 422.0.

Identify the significant figures of 0.0033

Leading zeros are not significant, they're just placeholders. Therefore, 0.0033 has two significant figures. No adjustment is necessary here because it already has less than four significant figures.

Identify the significant figures of 902.10

The zero in 902.10 is significant because it is a trapped zero (between nonzero digits). Therefore, 902.10 has five significant figures. To get four significant figures, this needs to be rounded off. Applying rounding rules, 902.10 becomes 902.1.

Identify the significant figures of 0.02173

The zeros in the front are placeholders while the zero after 7 is a significant figure, so 0.02173 has four significant figures. No adjustment is necessary here because it already has four significant figures.

Identify the significant figures of 7000

In the number 7000, without a decimal point, the trailing zeros are not significant. They are considered placeholders. So, 7000 only has one significant figure, which is 7. Therefore, no adjustment is necessary here because it's already in less than four digits.

Identify the significant figures of 7.02

In the number 7.02, both digits and the decimal point are significant, leading to three significant figures. No adjustment is required because it already has less than four significant figures.

Identify the significant figures of 67,000,000

In the number 67,000,000, the trailing zeros are not considered significant without a decimal point. Hence, there are only two significant figures in this number. There is no need to reduce it to four significant figures, as it already has fewer than four.

Key concepts

Rounding Rules

Rounding helps streamline numbers, especially when they're lengthy, by maintaining a required number of significant figures. This is crucial in scientific and mathematical contexts. The key rule to follow is: if the digit right after your required amount of significant figures is 5 or more, you should round up. Conversely, if it is less than 5, you should round down.

For example, suppose you have the number 3984.6, and you need to present it with four significant figures. You look at the next number (the 6) because 8 is your fourth significant figure here. Since 6 is greater than 5, you'll round the number up to 3985. This method ensures numbers remain as precise as required, adhering to conventional mathematical standards.

Non-Zero Digits

All non-zero digits are automatically considered significant figures. This means they are important in determining the precision of a number. Whether the number is in the hundreds, thousands, or farther, recognizing these digits is essential.

In the numbers 3984.6 and 422.04, each non-zero digit counts towards the significant figures. No matter their position in the number, each contributes to the precision and accuracy you're working to maintain. Non-zero digits are always users' guideposts to rounding or trimming numbers in any calculations or assessments.

Placeholder Zeros

Placeholder zeros can sometimes be misleading, as they aren't always significant. It's crucial to identify them right so that numbers are interpreted correctly. For instance, zeros in the number 0.0033 are not significant as they do not add to its precision, merely serving the role of placeholders to get you to the significant figures (3 and 3 in this case).

Meanwhile, trailing zeros can sometimes also act as placeholders, as seen in the number 7000. Without a decimal point, these trailing zeros are not considered significant. Their job is merely to indicate that the number isn't 7, but a value much larger, precisely understood through its position. This distinction is vital in various numeric evaluations, especially in scientific disciplines.

Decimal Point Significance

Decimal points play a crucial role in the significance of numbers. They not only impact the readability of numbers but also determine which zeros are significant. When a decimal is present, zeros trailing non-zero digits become significant, as they communicate precision.

For instance, in the number 902.10, the ending zero is significant only because of the decimal. It implies a higher precision than if the number were simply written as 902.1. Knowing how to interpret numbers with decimals correctly ensures that you maintain the intended level of detail and accuracy. Thus, decimals are pivotal in reflecting the exactness and quality of numerical data.