SCH 3U/4C - Metric Prefixes & Significant Digits

So now, the moment you've all been waiting for, let's get mathy in the house.  

Converting Between Metric Prefixes

Prefix Sym-bol	M	-	-	k	h	da	Base (g, s, L, m)	d	c	m	-	-	µ
Prefix Name	mega			kilo	hecto	deca	-	deci	centi	milli			micro
10x	106	105	104	103	102	101	100	10-1	10-2	10-3	10-4	10-5	10-6

Treat the table above like a modified number line.  Notice that, from left to right the values decrease by a factor of ten as you jump from box to box.  Also notice that not every box has a metric prefix associated with it - that's okay, those spots still represent a position on the "number line."

When converting, place your finger on the prefix with which you're starting and count as you hop from box to box, arriving at the prefix you desire.  The decimal point in the original number will hop in the same direction and the same number of times as your finger just did.

ex.  0.825 kL  =  _____ daL  (you hopped 2 spots to the right and the decimal point follows suit)

⇛ 0.825 kL  =  82.5 daL

ex.  427 000 g = _____ Mg (you hopped 6 spots to the left and the decimal point does the same)

⇛ 427 000 g = 0.427 Mg

ex.  5 kL = ______ L (you hopped 3 spots to the right, as does the decimal point - fill in empty spots with zeroes)

⇛ 5 kL = 5 000 L

***This video is useful if you can't pick up what I'm laying down.***

  

Scientific Notation

     Converting very large or very small numbers into something more easily written is the purpose of scientific notation.  To do this, move the decimal point until it arrives at a position where there is one non-zero value in front of it.  Then add a "x10^x" to the new number.  The exponent, x, will be positive if the decimal jumps to the left and negative if it jumps to the right.

 

     ex. 7 456.1  ⇛ 7.456 1 x 10³

     ex. 0.000 000 043 7 ⇛ 4.37 x 10^-8

Significant Digits (aka Significant Figures)

Significant Digits (SD) are useful in tracking the quality of a measurement.  Obviously, 123.012 is fancier than 123.  Sig digs are also useful when doing a calculation that requires an answer with appropriate rounding.  But how do we determine sig digs?

Rules for Determining Significant Digits

(a)   All digits from 1 through 9 are considered significant.

ex. 1 245 – 4 SD                237.877 – 6 SD          4.789 991 – 7 SD

(b)   Zeroes between significant digits are significant.

ex. 101 – 3 SD                  1.004 52 – 6 SD         102 400 601 – 9 SD

(c)   Zeroes to the left of significant digits, serving only to locate the decimal point, are not significant.

ex. 0.000 000 1 – 1SD; it can be rewritten as 1 x 10^-7 and now it’s obvious that there is 1 SD

(d)   Zeroes to the right of significant digits are only significant if they follow a decimal place.

ex. 1 500 – 2 SD               1.500 x 10³ – 4 SD     1.930 000 – 7 SD

(e)   Exact numbers, like the number of people in a room or the number of meters in a kilometer are considered to have an infinite number of significant digits.

Addition & Subtraction

When adding or subtracting numbers, the answer should have as many decimal places as the quantity in the problem with the least number of decimal places.

ex. 15.2345 + 0.11 + 45 = 60.344 5  

(which is rounded to 60 since the least number of decimal places in the data is zero and the digit after the 0 in 60 is less than＜5, so there is no need to round up)

Multiplication & Division

When multiplying or dividing numbers the answer should have as many significant digits as the quantity in the problem with the least number of significant digits.

ex. 321.4 ÷ 19 = 16.915 789 47 

(which is rounded to 17 since the least number of significant digits in the data is two and the digit after the 6 in 16 is ≧ 5, so the answer is rounded up)

***Here's another video that you may find helpful.***

****Failure to report final answers with the correct number of decimal places or sig digs (as appropriate to the question) will result in the loss of 0.5 mark per instance.  So, if you're having issues with any of this, seek help sooner, rather than later.****

Homework
1. Convert the following.
(a)  17 L = __________ mL
(b)  0.8 g = __________ mg
(c)  3.0 kg = __________ hg
(d)  0.023 kg = __________ g
(e)  549 000 mg = __________ g
(f)  8 000 mm = __________ µm
(g)  25.2 g = __________ mg
(h)  5 200 µs = __________ s
(i)  4.2 dm = __________ mm
(j)  21 mg = __________ g

2. "Scientific Notation & Significant Digits" handout
 

 And here are the answers:

Answers to Student Questions
Question 1: Why does 0.000 789 0 have 4 sig digs?

The 7, 8 and 9 are significant because of rule (a). 

The zeroes in front of 789 are not significant because they are simply place holders (rule (c)) to locate the decimal point in the proper position.  Notice that they would disappear if the number was put into scientific notation.  Therefore, those zeroes are not significant.

The zero after the 789 is significant because it follows a decimal point and a least one significant digit (rule(d)). You are right in saying that it doesn't have to be there, so because it is there, it must have been measured and is therefore important (or significant).



Question 2: In the "Addition & Subtraction" part of the lesson, why isn't 60.3445 rounded to 60?

Read that section of the blog again.  I said that 60.344 5 is, "rounded to 60 since the least number of decimal places in the data is zero and the digit after the 0 in 60 is less than＜5, so there is no need to round up."



Question 3: Can you go over some of the answer in #4?

Question 4: I still don't get why 4(a) is 800 when the calculator tells me 795.11.

795.11 needs to be rounded to 2 sig digs.  So, we keep the 7 and we keep the 9.  However, the next digit is 5, which rounds the 79 to 80.  So, now we have 80_.
We need something to put in the _ position to act as a place holder, so we put in a 0, giving us 800.