Significant Digits, Exact Numbers, Rounding and Operations

Let's say your're driving across the country in your Dodge Caravan, or Honda Civic if you want to promote the stereotype, and you come across a sign that says, "Entering Awesometown, population: 271,835". This figure is most likely COMPLETELY AND UTTERLY INCORRECT. But what if the sign said, "Entering Awesometown, population: 270,000". This figure is not entirely incorrect. By applying significant digits, this sign has become 68% more reliable.

A precise less precise number like 270,000 implies that the zeros are subject to change, whereas a number like 271,835 implies that the digits are there and cannot change and in most situations when dealing with inconsistent numbers like population, these precise numbers are incorrect.

HOWEVER, Significant digits or Significant figures or if you want to be super cool, Sig-Figs must be measured and have at least some certainty. The rule is that the last sig-fig in a number is uncertain while all other sig-figs are certain.

Ex. 5486; digits 5, 4 and 8 are certain while 6 is not

But how does one know which which digits are significant? Well, I have devised a set or rules, or commandments if you will for telling which digits are significant.

THE 5 COMMANDMENTS OF SIGNIFICANT DIGITS:

COMMANDMENT 1) All non-zero digits ARE significant.
ex. 7.342 has 4 Sig-Figs

COMMANDMENT 2) Zeros that trail after Sig-Figs and are before the decimal ARE NOT significant.
ex. 1340000 has 3 Sig-Figs

COMMANDMENT 3) Zeros that trail after the decimal point ARE significant.
ex. 54.00 has 4 Sig-Figs

COMMANDMENT 4) Zeros that come before Sig-Figs ARE NOT significant.
ex. 0.000078 has 2 Sig-Figs

COMMANDMENT 5) Zeros between other Sig-Figs are ARE significant.
ex. 5400.045 has 7 Sig-Figs

Exact Numbers


Some numbers have a set value assigned to certain things. One human has a set value of 1, if the subject is how many humans. We cannot say that there are 3.5 humans unless we were able to find 3 humans and half of one. If 3.5 humans were somehow calculated it would be rounded down to 3 humans.

Rounding

you will need to know your general place values, refer to this chart

If you have calculated an ugly answer such as 6542.5865 and you can only keep 2 significant digits, you must round your answer to accomplish this.

To do so, follow the basic rounding rules:

Look to the digit to the right of the desired place value that you want to round your answer to. If that number is larger than 5 round up and vice-versa. Simple, right? BUT WHAT IF THE NUMBER IS 5?!?!?!?!

CALM DOWN, BUDDY. First look to see if there are numbers after the 5. If so, round up because that indicates that the number is larger. If not, and the number is 5, this means that the number is exactly half-way, in which case, rounding it up wouldn't really be more correct or more incorrect than if you rounded it down. However, if you have a set of digits, you wouldn't want your data to be rounded up more than down. Round to the nearest even, or in other words, if the digit to the left of 5 is odd, round up, if it is even, round down.

Operations with Significant Digits

Addition and Subtraction:

line up your numbers like you would if you were doing old-school math, pre-calculator style with the decimals lined together.

    779.233

+   68.4    

    710.833

Examine the numbers: 779.233 goes to the thousandths place while 68.4 goes the the tenths place. ALWAYS PICK THE LOWER ONE. Round your answer to the tenths place. 710.8. Same rule applies for subtraction.

Multiplication and Division

44.56 * 846651.243 = 37726779.38808

Again examine your numbers and again, choose the lower one. This time, notice the number of sig-figs in each number. 44.56 has 4 while 846651.243 has 9. Round your answer to 4 sig-figs. 3773. Same rules applies for division.

SO THE CONCEPT IS: Lower amount of sig-figs means more reliable, because it reduces the chances of being COMPLETELY AND UTTERELY INCORRECT.

TL;DR: Watch this vid of Mr.Coolteacherguy with the pro hair talk about Sig-Figs.