Saturday, May 16, 2015

[S1_Expository] Significant Figures Made Easy

Introduction
     When you read a textbook on the topic of significant figures, you often get a bunch of complicated rules which seem pretty arbitrary and do not make sense.  Frankly, I too found these rules confusing.  They do not explain why those rules exist.  But there is sure-fire short-cut way to round a number to a given number of significant figures, based on understanding two key concepts: (1) the meaning of “significance” and (2) the role of ‘0’.  I present these below.

What does “significant” figure mean?
     The word “significant” simply means important.  Suppose you sold a sofa to somebody for
$7 287.  If the buyer wrote you a cheque for  $1 287,  there is an error in the leftmost first digit and you would get  $6 000  less.  That is bad.  If, instead, the buyer wrote you a cheque for  $7 281,  there is an error in the fourth digit and you would get  $6  less.  Bad, but not so bad.  Compare the 7 on the left against the 7 on the right.  Which digit is more significant?  As you can see, the digits on the left are more significant, and decrease in significance as we move towards the right.

The Roles of ‘0’ (Zero)
     Our modern Arabic-based place-value numerical system evolved over many centuries.  It is definitely far superior than Roman numerals.  But what is the symbol ‘0’ for?
     One usage of ‘0’ is to denote nothing, or the absence of something.  In Buddhism, which originated in India, there is this concept of sunyata (from Sanskrit), which means emptiness.  On a stone inscription in a Hindu temple at Gwalior in India, there is a small circle which stands for the digit zero.  Perhaps that was where ‘0’ originated from.  Is life an illusion?  Why then do we use something (a symbol ‘0’) to represent nothing?  Hmmmm .... Anyway, zero as a number is quite convenient to use in everyday life.  If you check up a website of a store for their inventory of jPhone VII mobile phones, and you see ‘0’, you know that they do not have that model of mobile phones in stock.  Without the number zero, they would have to make two separate lists: one for the models in stock and another for those out-of-stock.  By the way, all modern computing, storage and communication depends on 1’s and 0’s in various forms.
     Another usage of ‘0’ is that of a place-holder.  The ancient Chinese had a decimal place-value system for calculations in sand, but they used blank spaces on a grid instead of ‘0’s.  It is difficult to distinguish between say  507 (five hundred and seven)  and  5007 (five thousand and seven), especially if the grid was erased.  As you can see, ‘0’ is used to “hold the forte” in our decimal system.  Without the ‘0’,  507  and  5007  would both collapse to become  57, which means something entirely different.  Similarly, for the decimal  0.0009876,  the leading zeros function as place-holders.  The first significant figure is  9,  the second significant figure is  8,  and so on.  In general, placeholder ‘0’s are needed on the left of the decimal point, as well as to the right of the decimal point. 

Rounding to significant figures
     With the above understanding, here is the easy and sure-fire way to round to a certain number of significant figures:-
Step 1:  Skipping zeros on the left if necessary, identify the first non-zero digit.
Step 2:  From this digit, start highlighting the desired number of significant figures.
Step 3:  Look at the digit just after the last highlighted digit.  If this is 4 or below, do nothing.  If
              it is 5 and above, add one unit to the last highlighted digit.  Carry to the left if
              necessary.

I have highlighted the required significant figures (or digits) in yellow.  After we have identified the first non-zero digit, we just highlight the required number of significant digits, without caring about whether or not anyone of them is zero.  Then we just look at the digit after that to see if we need to round up.

Summary
     To round certain number to  n  significant figures:-
Step 1:  Identify the first non-zero digit from the left.
Step 2:  From this digit, start highlighting  n  significant figures.
Step 3:  Look at the digit just after the last highlighted digit.  Round up if
             this is 5 and above, otherwise do nothing.
Always remember to include the place-holder ‘0’s.

Suitable Levels
Lower Secondary Mathematics (usually secondary 1 ~ equivalent to about grade 7)
* other syllabuses that involve estimation and approximation




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