Significant Digits and Scientific Notation
Scientific Notation:
In science, math and computers you will come across some very large and some
very small numbers. For example, when we are discussing the universe we
know that the distance from the Earth to the Sun is 152, 000, 000, 000 m. When
we are in chemistry the number of atoms in a mole of substance is given by
Avogadro’s number which is 602,200,000,000,000,000,000,000. On the other
side of the scale in physics the mass of an electron is quoted as
0.00000000000000000000000000000911 kg.
The point is that these numbers can be very tedious and time consuming to write
out in this way. Therefore scientists developed a kind of shorthand for writing
these numbers known as Scientific Notation. This shorthand allows us to omit
the enormous number of zeros in these large and small numbers.
Method
1. Start after the first nonzero digit from the left.
2. Count how many places you move to reach the end
3. Write down the number with all of the nonzero digits and add a X 10N,
where N is the number your counted from step 2
4. If you moved the decimal to the left N is negative, if you moved it to the
right N is positive
Example: 0.00000000000000000000000000000911
1. 9.11 X 10-31 [Since I had to count to the left]
Example: 602,200,000,000,000,000,000,000
1. 6.022 X 1023
Significant Digits
When we complete calculations with measured numbers or empirical data we
must be careful when reporting the calculated value. Consider the following
example. Suppose you wish to find the area of your desk with a millimetre ruler.
You measure the length to be 50.43 cm and the width to be 75.45 cm. Then the
area is given by the length times the width which yields a value of 3804.9435 cm 2.
Is this an acceptable value for the area of the desk? Can you really be confident
with the precision of this calculation? The answer is no!! It is impossible to
measure lengths that are accurate to two decimal places and then expect the
area to be accurate to four decimal places. In these types of situations we must
take into account significant digits.
Identifying Significant Digits
1. All non – zero digits are significant digits
2. Zeros appearing between two non zero digits are significant
3. All zeros appearing to the right of a decimal point and non zero digits are
significant
4. All zeros appearing in a number without a decimal point to the right of the
last non zero number ARE NOT significant
5. All zeros appearing to the right of a decimal point but before non zero
numbers ARE NOT significant
NOTE: Zeros like those discussed in 4 and 5 are merely place holders for the
number. We can write the number using scientific notation without including
these zeros.
Examples:
1.
2.
3.
4.
134.45
This number has 5 significant digits
120.403
This number has 6 significant digits
125.300
This number has 6 significant digits
10340000
This number has 4 significant digits the last four zeros are
place holders and this number could be written as 1.034 X 10 7.
5. 0.00000234 This number has 3 significant digits since the first five zeros
are place holders and this number could be written as 2.34 X 10 -6.
Arithmetic with Significant Digits
In order to correctly express calculated number with the appropriate amount of
significant digits we must adhere to the following rule.
Rule of Thumb for Significant Digits
When completing calculation with measured numbers complete the calculations
as normal and then round to the smallest number of significant digits in the
numbers used. Apply scientific notation as necessary.
Examples:
1. Calculate the area of the round table whose diameter is 10.5 cm long.
d
10.5
r 
 r  5.25
2
2
A   r 2  A  (3.14159)(5.25)  A  16.4933475
r
But the number we used in the calculation only had 3 significant digits therefore
the area may only have 3 significant digits. Therefore our answer must be
rounded to A = 16.5 cm2.
2. Calculate the perimeter of the triangle whose sides measure 14.3 cm, 12.30
cm and 10 cm.
P = 14.3 + 12.30 + 10  P = 26.60 cm
However one of the numbers only has one significant digit so we must round our
answer to P = 30 cm.
3. The energy of an object is given by the formula E = mc2, where c represents
the speed of light taken to be 3.00 X 108 m/s. Calculate the energy possessed
by an electron whose mass is given by 9.11 X 10-31.
E = (9.11 X 10-31)(3.00 X 108) = 2.733 X 10-22 J
We must round to three significant digits, therefore the energy is given by
E = 2.73 X 10-22 J.