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For each of the following numbers, indicate which zeros are significant and explain. Do not merely cite the rule that applies, but explain the rule. a. 10.020 b. 0.002050 c. 190 d. 270

Short Answer

Expert verified
In the given numbers: a) In 10.020, both zeros are significant; one lies between non-zero digits and the other is at the end of the decimal indicating the level of accuracy. b) In 0.002050, two zeros are significant (between 2 and 5, and after 5); the first two zeros are not significant as they just serve as placeholders. c) In 190, the zero between 1 and 9 is significant, as it distinguishes it from 19 and denotes accuracy. d) In 270, the zero after 7 is not considered significant as it is a trailing 0 without a decimal.

Step by step solution

01

a. 10.020 - Identifying significant zeros

: In the number 10.020, there are two zeros that we need to consider. Both zeros are significant in this case. The zero between the 1 and 2 is significant because it is a non-zero digit, and zeros between non-zero digits are always significant. The last zero after the 2 is also significant because it is at the end of the decimal number, and zeros at the end of a decimal number are considered significant as they indicate the level of accuracy and precision. So, both zeros in 10.020 are significant.
02

b. 0.002050 - Identifying significant zeros

: In the number 0.002050, all the zeros are considered. The first two zeros after the decimal point (0.00) are not significant because they are zeros preceding the first non-zero digit and serve only as placeholders. The next zero between 2 and 5 is significant, as it is a zero between non-zero digits. The last zero after the 5 is significant because it comes at the end of the decimal number, and again, indicates the level of accuracy and precision. So, in 0.002050, two zeros are significant (the one between 2 and 5 and the one after the 5).
03

c. 190 - Identifying significant zeros

: For the number 190, there is one zero, located between the 1 and the 9. This zero is significant because it is in-between non-zero digits, and it shows the difference between 19 and 190, which have different values and have different levels of accuracy.
04

d. 270 - Identifying significant zeros

: In the number 270, there is one zero, located after the 7. Since there is no decimal point, it is uncertain if the zero is significant. However, often when a number is written without a decimal, it is assumed the trailing zeros are not significant. To make it clear that a zero in this case is significant, 270 would need to be written as 270. or 2.70 x 10^2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Measurement Precision
Measurement precision in chemistry is crucial as it reflects the exactness of a measurement or how close repeated measurements are to each other. The concept of significant figures is directly linked to precision since it indicates which digits in a number are meaningful and contribute to its accuracy.

For instance, the number 10.020 demonstrates a precision to the thousandths place, signifying that the measurement was precise enough to warrant three decimal places. Understanding the precision of measurements allows chemists to convey the certainty of their findings accurately and helps in identifying potential variances and the reliability of the data.
Identifying Significant Zeros
Identifying significant zeros can be challenging, but knowing a few simple rules can provide clarity. Zeros play a critical role in showing the scale and precision of a number, and their significance can be determined by their location.

For example, in the decimal number 0.002050, the first two zeros are merely placeholders to express the small magnitude of the number and therefore are not significant. However, the zeros between the non-zero digits (5 and 2) and the zero at the end of the number after a non-zero digit (5) signify precision and are counted as significant figures. This distinction is essential in chemistry where the difference between 0.00205 and 0.002050 could imply a large discrepancy in measurements.
Scientific Notation
Scientific notation is a method of writing very large or very small numbers in a concise and standardized form. It features two parts: a coefficient (a number between 1 and 10) and a power of ten, which expresses how many times the coefficient needs to be multiplied by ten to reach the original number.

For instance, 270 can be uncertain in regards to significant figures. Writing it as 2.70 x 102 in scientific notation clarifies that all digits are significant, conveying precision that regular notation may fail to communicate. Especially in chemistry, where measurements at the atomic or molecular scale are common, using scientific notation can help clearly and accurately represent quantities.

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