Chapter 2

Multiplication in scientific notation

1) Multiply the coefficients (the decimal numbers between 1 and 10) together in the usual manner. 2) Add algebraically the exponents of the powers of ten to obtain a new exponent.

Rules for significant digits

1) non-zero digits are always significant 2) leading zeros are never significant 3) confined zeros (in between nonzero digits) are always significant 4) trailing zeros are sometimes significant a. when there is a decimal point present in the number b) if they carry overbars

decimal to scientific tip

Movement to the left makes the exponent positive, and movement to the right makes the exponent negative.

Operational rules (multiplication and division)

Significant figures as considered. The number of significant figures in the product or quotient is the same as in the number in the calculation that contains the "fewest significant figures." Example: 6.038 x 2.57 = 15.51766 which becomes 15.5 because the least number of significant figures in this operation is 3.

Uncertainty considerations for calculator answers of addition/subtraction of scientific notation problems

The answer should have the same uncertainty as the greater uncertainty of the of original problem. Thus, adjusting the significant figures accordingly.

Measurement

The chemist's most used tool. it is the determination of the dimensions, capacity, quantity, or extent of something.

Two types of information must be conveyed whenever a numerical value for a measurement is recorded

1) the magnitude of the measurement 2) the uncertainty of the measurement

Converting from scientific to decimal notation

Examine the exponent, then move the decimal point accordingly; if positive, move decimal to the right; if negative, move decimal to the left. Number of sig figs must be the same in decimal and scientific form.

Addition or subtraction in scientific notation

For addition/subtraction to take place the power of ten for all numbers must be the same. If you need to adjust the power of ten, do so. The adjusted number will be greater than or less than ten, but this is okay during the process. Add/subtract the coefficients and the exponent is maintained at its now common value. Changing the smaller exponent to a larger one will usually produce a coefficient in the answer that is a number between 1 and 10.

The precision and accuracy of measurements usually relate directly to the actual measuring device used

For example, the precision and accuracy of of temperature readings obtained from a thermometer with a scale marked in tenths of a degree would be better than readings obtained from a thermometer whose scale has only degree marks. Errors in measurement can be classified as either systematic errors or random errors.

Two most common origins for uncertainty in a measurement

Human error: our hands have limits when operating equipment, and our eyes can see only so well. There are limits to how well a person can calibrate an instrument. Instrument error: Imperfections of an instrument, or wear and tear.

Uncertainty and scientific notation

The uncertainty associated with a scientific notation number is obtained by determining the uncertainty associated with the coefficient and the multiplying with the coefficient and then multiplying this value by the exponential term. For example, the number 3.753 x 10^2 10^-3 x 10^2 = 10^-1 uncertainty exponential uncertainty of coefficient term in value 10^-3 is in the tenths place of the coefficient. Times this by the exponential term 10^(-3+2=-1) you can see that 10^-1 is the uncertainty when you write number in decimal notation = 375.3 the uncertainty is in the tenths place.

Scientific notation

a numerical system in which numbers are expressed in the form A x10^n, where A is a number (0<A<10), and n is a whole number. A is the coefficient, and 10^n is the exponential term. Uses 10 as common base; its decimal equivalent is the number 1 followed by as many zeros as the power. (Example: 10^2 =100, 10^4=10,000) when negative exponent the absolute value of the power is always one more than the number or zeros between the decimal point and the one. (Example: 10^-2=0.01, 10^-4=0.0001

Order of magnitude

a single exponential value of the number 10. Example: 10^6 is four orders of magnitude larger than 10^2, and 10^7 is three orders of magnitude larger than 10^4.

Accuracy

an indicator of how close a measurement (or the average of multiple measurements) comes to a true or accepted value (true value: most accurate currently known value for a measured quantity).

Precision

an indicator of how close a series of measurements on the same object are to each other. Key word: series.

Significant figures

are the digits in any measurement that are known with certainty plus one digit that is uncertain

It is possible to obtain measurements based on averaging in which high random error is present

if the "highs and lows" balance each other to produce an average that is close to the true measurement value.

Systematic error

is a "constant" error associated with an experimental system itself. For example, incorrect calibration of instrument; a flaw in a piece of equipment; or presence of an interfering substance within the experimental system. Usually affects the accuracy, and not the precision, of a measurement.

Inexact number

is a number that has a value with a degree of uncertainty in it. Accompanies a measurement, always.

Exact number

is a number that has a value with no uncertainty in it; that is, it is known exactly. (for example: there are 12 objects in a dozen, not 12.01, or 12.02)

Random error

is error caused by uncontrollable variables in a experiment. For example, changes in air currents or temperature near a sensitive instrument. Precise measurements are an indication of minimal random error.

The magnitude of the uncertainty in the last recorded digit in a measurement (estimated digit)

may be indicated by a plus-minus notation. For example, 15 "plus-minus" 1 gram 15.3 "plus-minus" 0.1 gram 15.34 "plus-minus" 0.03 gram

Accurate measurements have low

systematic error and generally random error.

Uncertainty is always present in measurements

the last digit of any measurement is estimated, and it is this estimated digit that reflects uncertainty. The more decimal places there are the less uncertainty there is. For example, 29.2 has more uncertainty than 29.25. Only one estimated digit is ever recorded as part of a measurement .

Rounding off

the process of deleting unwanted (nonsignificant) digits from a calculated number. Three simple rules govern the process.

Recording measurements to the proper uncertainty level

the recorded uncertainty is smaller, by a factor of 10, than the scale unit interval. examples: a) a thermometer scale with markings in 1 degree intervals--- plus-minus 0.1 degree b) a measuring cup scale with markings in 100 fluid ounce intervals -- plus-minus 10 fluid ounces c) a ruler scale with markings in 10 centimeter intervals -- plus-minus 1 centimeter d) a barometer scale with markings in 0.1 of a millimeter intervals -- 0.01 You just basically divide the scale unit interval

Division in scientific notation

1) Divide the coefficient (the decimal numbers between 1 and 10) in the usual manner. 2) Subtract algebraically the exponent in the denominator from the exponent in the numerator to give the exponent of the new power of ten.

Converting from decimal to scientific notation

1) The coefficient must be a number between 1 and 10 that contains the same number of significant figures as are present in the original decimal number. The coefficient is obtained by rewriting the decimal number with a decimal point after the first nonzero digit and deleting all nonsignificant zeros. 2) The value of the exponent for the power of ten is obtained by counting the number of places the decimal point in the coefficient must be moved to give back the original decimal number. If the decimal point movement is to the right, the exponent has a positive value and if the decimal point movement is to the left, the exponent has a negative value. numbers >1 will always have positive exponents, and numbers <1 will always have negative exponents.

When both multiplication/division and addition/subtraction occur in the same problem

1) The intermediate answer is rounded such that one extra digit is maintained (the first insignificant digit), which is highlighted in a way to denote that it is not significant. 2) This "incorrectly" rounded intermediate answer (one too many digits) is then carried into further steps of the calculation and "correct" rounding occurs with the final answer.

Rules of rounding off

1) if the first digit to be dropped is less than 5, that digit and all digits that follow it are simply dropped. (ex. 62.312 is 62.3) 2) if the first digit to be dropped is a digit greater than 5, or a 5 followed by digits other than all zeros, the excess digits are all dropped and the last retained digit is increased in value by one unit. (ex. 62.782 is 62.8) 3) if the fist digit to be dropped is a 5 not followed by any other digit or a 5 followed only by zeros, an odd-even rule applies. Drop the 5 and any zeros that follow it and then a. increase the last retained digit by one unit if it is odd, or b. leave the retained digit the same if it is even. (example: 62.650 becomes 62.6--even rule, or 62.350 becomes 62.4--odd rule)

Significant figures and exact numbers

They will have no effect on the considerations of significant figures. They are thought to have an infinite number of significant figures.

Significant figure multiplied by an exact number

Two approaches: 1) Treating the problem as a simple multiplication problem using the multiplication/division significant figure rule when rounding the calculator answer. 2) Treating the problem as an addition problem in which the measured quantity is added to itself several times and then rounding the calculator answer using the addition/subtraction rule

Operational rules (addition and subtraction)

Uncertainties are considered in addition and subtraction of series of statements, the uncertainty in the answer should be the same as that of the measurement in the series that has the "greatest uncertainty." Example: position of uncertainty 347 ones position + 2.03 hundredths position + 23.6 tenths position --------------- 372.63 (calculator answer) 373 (correct answer)