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 surefire shortcut 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 Arabicbased
placevalue 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
outofstock. 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 placeholder. The ancient Chinese had a decimal placevalue
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 placeholders. 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 surefire way to round to a
certain number of significant figures:
Step 1: Skipping zeros on the left if
necessary, identify the first nonzero 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.
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 nonzero 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 nonzero 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 placeholder ‘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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