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Chapter 1: Problem 13

Determine the number of significant digits in each of the given approximate numbers. $$6.80 ; 6.08 ; 0.068$$

Short Answer

Expert verified
6.80 has 3 significant digits, 6.08 has 3 significant digits, and 0.068 has 2 significant digits.

Step by step solution

01

Understanding Significant Digits

Significant digits (or significant figures) in a number include all the digits that are known with certainty plus one last digit that is estimated. When determining significant digits, we pay attention to non-zero digits, any zeros between significant digits, and any trailing zeros in a decimal number.
02

Analyzing 6.80

For the number 6.80, we start from the first non-zero digit, which is 6 and then included the subsequent digits. The digit 8 and the zero (0) following it are considered significant because trailing zeros in a decimal number are considered significant. Hence, 6.80 has 3 significant digits.
03

Analyzing 6.08

In the number 6.08, the digits 6, 0, and 8 are all significant. The zero is significant because it occurs between two significant figures, 6 and 8. Therefore, 6.08 has 3 significant digits.
04

Analyzing 0.068

For the number 0.068, the zeros to the left of non-zero digits are not significant. The digits 6 and 8 are significant. The zero between 6 and 8 is also significant as it lies between significant figures. Therefore, 0.068 has 2 significant digits.

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Key Concepts

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

Non-Zero Digits
In the realm of significant digits, non-zero digits play a primary role. Non-zero digits are essentially any number other than zero (1 through 9), and they are always considered significant. These digits are crucial because they represent the actual quantity of a value.

  • For example, in the number 6.08, each digit鈥�6 and 8鈥攁re significant because they are non-zero.
  • They form the backbone of our understanding of significant figures.
The perspective shifts slightly with how zeros are treated, especially when they appear in various positions and contexts, such as within trailing zeros or between non-zero digits.
Trailing Zeros
Trailing zeros are zeros that appear at the end of a number after a decimal point. In the context of significant digits, trailing zeros assume importance only when they appear in a decimal number, not necessarily when they are in whole numbers.

For instance, 6.80 has trailing zeros, and because it is a decimal number, these zeros are significant. Principles for identifying significant trailing zeros include:
  • Trailing zeros in a decimal number (after the decimal point) are significant.
  • In 6.80, the zero after 8 gives precision, so it is counted as a significant digit.
This accuracy is important for making precise measurements and computations.
Decimal Numbers
Decimal numbers consist of digits both before and after a decimal point. Importantly, the decimal point helps define which zeros are significant.

In a number like 0.068:
  • The digits 6 and 8 are significant because they are non-zero.
  • The zero between these two digits is significant because it is nestled between significant digits.
  • However, zeros that precede the first non-zero digit after the decimal point are not considered significant, hence the initial zeros in 0.068.
Understanding how these numbers work ensures that you properly interpret and utilize significant digits in your calculations.

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Most popular questions from this chapter

Perform the indicated multiplications. The weekly revenue \(R\) (in dollars) of a flash drive manufacturer is given by \(R=x p,\) where \(x\) is the number of flash drives sold each week and \(p\) is the price (in dollars). If the price is given by the demand equation \(p=30-0.01 x,\) express the revenue in terms of \(x\) and simplify.

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. A flare is shot up from the top of a tower. Distances above the flare gun are positive and those below it are negative. After 5 s the vertical distance (in \(\mathrm{ft}\) ) of the flare from the flare gun is found by evaluating \((70)(5)+(-16)(25) .\) Find this distance.

A jet travels 600 mi/h relative to the air. The wind is blowing at \(50 \mathrm{mi} / \mathrm{h} .\) If the jet travels with the wind for \(3 \mathrm{h},\) set up the expression for the distance traveled. What fundamental law of algebra is illustrated?

In blending two gasolines of different octanes, in order to find the number \(n\) of gallons of one octane needed, the equation \(0.14 n+0.06(2000-n)=0.09(2000)\) is used. Find \(n,\) given that 0.06 and 0.09 are exact and the first zero of 2000 is significant.

Solve the given problems. Refer to Appendix B for units of measurement and their symbols. A baseball player's batting average (total number of hits divided by total number of at-bats) is expressed in decimal form from 0.000 (no hits for all at-bats) to 1.000 (one hit for each at-bat). A player's batting average is often shown as 0.000 before the first at-bat of the season. Is this a correct batting average? Explain.
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