Lecture 18 – More Tips on Significant Figures
When multiplying or dividing numbers, the significant digits of the number with the least number of digits gives the number of significant digits the answer will have.
Example: 40 lbs. potatoes X $0.45 per lb. equals $18.00 or $18 since the first number is only measured to two places. That was easy, right?
Example: 0.5 ounces of perfume X $25.00 per ounce equals $12.50 for 0.5 ounces of perfume. (Note: the zero is only written because you cannot divide the choice further.)
Example: 6.23 ft of wood X $2.00 per linear foot equals $12.50 per linear foot.
Imagine you are attempting to prove a theory based on a specific property, like boiling point. Unless the boiling point temperature was recorded precisely by other chemist, you would have trouble repeating this experiment. It is ridiculous to feel that you could prove a new theory, since theories become laws by repeated experimentation. Therefore, we must record measurements precisely.
Scientific knowledge moves forward by building upon results and experiments performed by earlier scientists. If measurements are taken carelessly, scientists do not know if the results are new and promising or just incorrect.
Precision – the closeness of the two sets of measured groups of values.
Precision is directly related to the amount of reproducibility of a measurement. Closely related to the topic of precision is that of accuracy. Some people use the two interchangeably, but that is incorrect. There is a difference.
Accuracy – is linked to how close a single measurement is to its true value.
In baseball, Player 1 throws balls at a target’s center, it represents high precision and accuracy. Player 2’s aim with balls high and low missing the target, represents low precision and low accuracy. Player 3’s hits, clumped together at the bottom left side of the target, define high precision (since every ball landed in the same place, but low accuracy since the object was supposed to hit the target’s center.) Player 3, then, has to work on hitting the target’s center, if he wants to win games and improve accuracy.
Rounding is the way to drop non-significant numbers in a calculation and adjusting the last number up or down. There are three basic rules to remember when rounding numbers.
a) If a digit is greater than or equal to 5 followed by non-zeros, then add 1 to the last digit. (3.2151 would be rounded to 3.22.)
b) If a digit is less than 5 then the digits would be dropped. (7.12132 would be rounded to 7.12.)
c) If the number is 5 (or 5 and a bunch of zeros), then add 1 to the last digit. (4.825, 4.82500, 4.81500 all round to 4.82.)
Note: Rounding reduces accuracy, but increases precision. The numbers get closer, but not necessarily on target.
Examples: Round the following numbers for practice.
1) 2.2751 to 3 significant digits
2) 4.114 to 3 significant digits
3) 3.177 to 2 significant digits
4) 5.99 to 1 significant digits
5) 2.213 to 2 significant digits
6) 0.0639 to 2 significant digits
You should have arrived at 2.28, 4.11, 3.2, 6, 2.2, and 0.064 respectively.
Until you can do this effortlessly, have the rules right beside you. Read the rules aloud several times slowly. Write each rule on its own 3X5 card. Yard sales and thrift stores have old math workbooks that can be purchased for less than $1.00. In your leisure time, work a few math problems instead of turning on the 52-inch box of reality madness. You will be surprised at how enjoyable math can be. Make your children and grandchildren work a few problems, too. These exercises will also drive dementia and Alzheimer’s disease away.
Remember: When multiplying or dividing measurements, the number of significant digits of the measurement with the least number of significant digits, determines the number of significant digits in the answer.
Do you thoroughly understand how significant digits are figured out?
a) 1.8 pounds of oranges X $3.99 per pound equals $7.182 or $7.18 or $7.2
(Note: 1.8 pounds of oranges has two significant digits.)
b) 15.2 ounces of olive oil X $1.35 per ounce equals $20.50.
c) 25 linear feet of rope X $3.60 per linear foot equals $90.00.
Measurements can be calculated to high precision. Calculators give between 8 and 10 numbers in response to the numbers entered for a calculation, but most measurements require far less accuracy. Rounding makes numbers easier to work with and to remember. How would an out-of-town guest feel if you told her to drive 3.3334557 miles north on Route 1, take the jug handle onto Harrison Street and drive 3.445577 miles until you reach Walnut Street. Turn left on Walnut Street and drive 1.1110554 miles. The last red brick house on the left is mine! Well, your guest might never make it. The odometer is not even that meticulous.
Reference: Math examples were taken from Chemistry Demystified: A Self-Teaching Guide
by Linda Williams
As much as we all feel that we are well educated in America, we all have gaps in our knowledge. I have always earned A’s in math, but had a weird feeling in my stomach all the while. Subconsciously, I had internalized the negative conversations about girls lacking good spatial orientation. Once I identified and confronted the stereotype, this Black girl became the most confident math and science student ever. Did you get the point? I was making all A’s but still felt like I was just skating by. I reject ALL stereotypes about so-called ‘minorities’. I despise the word and am not a part of any minority group. I use the term diversity or diverse populations. Thanks to Dr. Molefi K. Asante and staff in the African-American Studies department at Temple University, I learned the negative connotation associated with terms like minority, fair-skinned, and non-white. In your practice, you will encounter people of every race, culture, and creed, and you will need to know how to respond to their needs with sensitivity and intelligence. My lectures run deep and wide, as my philosophy is that all disciplines eventually intersect.
In my lectures and labs, you will encounter anatomy and physiology, microbiology, biochemistry, pharmacology, and all of the other medical sciences. They will be presented in a way that is simplistic, detailed, and sometimes humorous. You can do this despite your age, race, or financial status. If I am your medical mentor, all you need is a willingness to learn and a lot of energy. Indolence won’t work with my programs. My students say, “Do not go to her if you are not ready. She is a perfectionist and hates to waste time.” If you approach me with a goal, I will push you until you accomplish it. Do well with this information. Also, eat lots of fresh fruit and vegetables, rest, exercise, and drink water.