Lecture 3:
Scientific Figures/Scientific Notation

A. Precision and Accuracy
B. Significant Figures
C. Calculations and Significant Figures
D. Scientific Notation

Measurement:

Why are measurements important?

1. Measurements are essential to our daily activities.
2. Measurements are essential to chemistry and biological sciences.
3. Measurements involve units such as feet (length), acres (area), gallons (volume), miles (distance), and pounds (weight).
4. Units are the language of measurement.
5. Making unit conversions require some basic arithmetic.

A. Precision and Accuracy

Precision and Accuracy are terms for quality of measurements. They have different meanings when used for scientific measurements.

Precision: how near repeated measurements of the same quantity agree with each other.

Example: If the repeated measurements of the same quantity agree with each other, the measurements have a high degree of precision.

Accuracy: how a measurement or multiple measurements agree with a true value (standard control).

In any science measurements, we usually strive for precision and accuracy. Precision depends on the fineness of calibration of the measuring device.

Example: A stopwatch versus a wristwatch. A stopwatch measures time more precisely than a wristwatch. Why?

1. The difference that can be read on the stopwatch is about 0.01 seconds.

2. The difference that can be read on the wristwatch is 0.5 seconds.

Errors in measurements can be caused by:

2. Choosing the wrong measuring device.

B. Significant Figures

The correct use of significant figures is essential in reporting any scientific measurements carried out in the lab. It is important to not that all digits obtained by measurement are significant. It is also important to know that the last digit to the right is an estimate.

To read more about significant figures, click here. There are several pages of this site, so be sure and click on Next at the bottom of each page.

C. Calculations and Significant Figures

Instructions for multiplying measurements:

1. First separate the numbers from the units.
2. Multiply the numbers on a calculator.
3. The units must also be multiplied.
4. When both the numbers and the units have been multiplied, the answer should be obtained.
1. Note: Suppose we want to find the area of a sheet of paper and we measured the length to be 20.14 inches and the width to be 9.45 inches.

Area = (length) (width)
Area = 20.14 in. × 9.45 in.
Area = 190.323 square inches

Question: Are we justified in reporting this six-digit number?
Question: Why?

1. Because our measurements of width and length contained only 4 and 3 significant figures respectively.
2. Since the least precise figure in our multiplication contained only 3 significant figures, our answer must have only 3 significant figures.
3. The correct value of the area is 190.00 square inches.

Note: Zeros used to show where a decimal point belongs are not significant.

Rules to ensure that your answers always contain the correct number of significant figures:

1. In multiplication and division, the answer must contain the same number of significant figures as the term with the least number of significant figures.

Example: (16.79) (14.6) = 245.134. The answers should be reported as 245.00. Why? Because the term with the least number of significant figures = 14.6 has only 3 significant figures.

2. In addition and subtraction, the answer must contain the same number of decimal places as the term with the least number of decimal places.

Example: 18.02
12.2
328.445
358.665
1. How will the answer be reported? The term with the least number of decimal places (12.2) has only 1 decimal place. The answer then can contain only one decimal place. Answer = 358.7

How to round off properly:

1. If the nonsignificant figure is less than 5, simply drop the nonsignificant figure.

Example: 6.243 = 6.24 when rounded off to three significant figures.

2. If the nonsignificant figure is more than 5, drop the nonsignificant figure and increase the significant preceding it by 1.

Example: 4.487 = 4.49 when rounded off to three significant figures.

D. Scientific Notation

A scientific notation number has the general formula: N  × 10exponent ; where N = a number between 1 and 10 and the exponent is a whole number that is the power to which a number is raised.

Example 1:

1,000,000 = 106
6 = the exponent
10 = the base

Example 2:

1,000,000 = 1 × 106
This is written by moving the decimal point six (6) places to the left and using the exponent 6.
1 million = 1,000,000 = 1 × 106

Types of Exponents

1. Positive exponent: indicates how many times a base must be multiplied by itself to produce the original figure.

Example:
1 million = 1,000,000 = (10)(10)(10)(10)(10)(10) = 1 × 106
A positive exponent signifies that the number is greater than 1.

2. Negative exponent: indicates how many times 1 must be divided by the base to produce the original figure.

Example:
1 million = 1/1,000,000 = 1/(10)(10)(10)(10)(10)(10) = 1 × 10-6
A negative exponent signifies that the number is less than 1.

How to switch from scientific notation to ordinary figures:

• Reverse the operation just described.
• A positive exponent indicates that the decimal point must be moved to the right to give the ordinary number.
• A negative exponent indicates the decimal point must be moved to the left to give the ordinary number.

You will need a scientific calculator for calculations involving numbers in scientific notation. The following keys must be used:

EE, EXP, or EEX (on the calculator)

Example: To enter the numbers 5.74  × 104

1. First enter 5.74
2. Press the EE, EXP, or EEX key
3. Press 4
4. The calculator will display:
5.74 × 1004
5. Then proceed with any calculations with this number on the calculator.

Negative numbers can be entered into the calculator by following the steps below:

1. Enter the number, for example, -0.000057, enter 5.7
2. Press the EE, EXP, or EEX key
3. Press the +/- key
4. Press 5
The number 5.7 × 10-05 will appear on the screen and is ready for further calculations.

Try these for practice:

Round off to 2 significant digits

1. 80.992
2. 0.006685
3. 2.2222

Positive Scientific Notation

1. 148,000 (3 significant figures)
2. 4,731 (2 significant figures)
3. 144,800 (2 significant figures)

Negative Scientific Notation

1. 0.0004714 (3 significant figures)
2. 0.0863 (3 significant figures)
3. 0.000000259 (2 significant figures)