6.10x10⁵ 
6.00
0.596
0.000610 

• The most significant digit is the left-most non-zero digit.  In other words, zeroes to the left are never significant.
• If there is no decimal point explicitly given, the right-most non-zero digit is the least significant digit.
• If a decimal point is explicitly given, the right-most digit is the least significant digit, regardless of whether it is zero or non-zero.
• The number of significant digits is found by counting the places from the most significant to the least significant digit.

3456
134700
0.003043
0.01000
1030.
1.057
0.0002307     (for more try this game)

As illustrated in this example, zeroes can be significant figures, and it is important to show them if so. Non-significant zeroes are most conveniently taken into account by using powers of ten (scientific notation). For example, if you wish to write 49000 to three significant figures, you should write:

4.90 x 10⁴

or writing -0.00034000 to three sig. figs would be:

-3.40 x 10^-4

Now we will look at how to combine numbers with differing amounts significant figures. In all cases the rule to follow is the number that makes the greatest contribution to the uncertainty of a result determines the amount of significant figures. 

• If the digit to be rounded off is below 5, the last significant digit remains unchanged.

• If the digit to be rounded off is above 5, the last significant digit should be raised by one.

• If the digit to be rounded off is exactly 5, the last significant digit should be raised by one only if it makes it an even number. 

Example: It is desired to determine the area of a rectangle whose sides have been measured and found to be 65.27 cm and 73.83 cm.  To determine the area, the product of the lengths of the two sides is taken which yields:

(66.27 cm) (73.83 cm) = 4818.8841 cm²

4819 cm²

(65.265 cm) (73.83 cm) = 4818.514950 cm²

 (65.275 cm) (73.83 cm) = 4819.253250 cm²

These areas differ from each other in the fourth place, and so there is no significance to be attached to the digits beyond the fourth place.  Therefore, only the first four places in the product are retained in this example.

Example 1, addition:

110.2  + 23.6632  + 0.0114 = 133.8746, and since 110.2 is the most uncertain (plus or minus 0.05) the answer must be rounded to 133.9 (also plus or minus 0.05). Note that 133.8746 implies we know it to plus or minus 0.00005!

Example 2, subtraction:

182.65 - 181.1 = 1.55 and since 181.1 is the most uncertain (plus or minus 0.05) the answer must be rounded to 1.6 (also plus or minus 0.05). 

Note that with addition or subtraction, the amount of significant figures in the result may be less that the amount in either of the inputs. In this case the answer has only two sig. figs, while the input numbers had 5 and 4 sig. figs respectively. 

(182.65 g - 181.1g)/(0.1402 cm³) = (1.55 g)/(0.1402 cm³)= 11 g/cm³

In this example the subtraction first gives a result with two sig. figs (but we do not round until the very end!) so we write in 1.55 g. We then compute the division, which gives 11.0556 g/cm³. Recalling that the subtraction left us with only two sig. figs, whereas the denominator has 4 sig figs, the final result must be rounded to two sig. figs or 11 g/cm³. 

Examples of exact numbers are;

• cardinal numbers: 4 apples, 8 marbles, 27 pages, a dozen eggs, etc...
• conversions: 12 inches = 1 foot, 100cm = 1 meter, 6.022 x 10²³ = 1 mole, etc...

"That's OK," replied the student. "It's important." He knew his friend has paid $35, and he had been given $50 out of the school's research budget to get the job done.

He returned the next day, expecting the job to be done. "Sorry," said the foreman. "We're still working on it. Try next week." Finally the day came, and our friend got his cube. It looked very, very smooth and shiny and beautiful in its velvet case. Seeing it, our hero had a premonition of disaster and became a bit nervous. He summoned up enough courage to ask for the bill. "$500, and cheap at the price. We had a terrific job getting it right -- had to make three before we got one right."

"But--but--my friend paid only $35 for the same thing!"

"No. He wanted a cube 2.1 cm on an edge, and your specifications called for 2.097. We had yours roughed out to 2.1 that very afternoon, but it was the precision grinding and lapping to get it down to 2.097 which took so long and cost the big money. The first one we made was 2.089 on one edge when we got finished, so we had to scrap it. The second was closer, but still not what you specified. That's why the three tries."

"Oh!"

Entering in too few significant figures may result in a lost point. The program will give you a full point for a problem even if you are off the expected result by 2%. For example, if the expected (correct) answer for a problem is 29.4 cm/s you would receive full credit for 29.6 cm/s, but you would lose that point for an answer of 30 cm/s since that is more than 2% over the actual value. 

Sometimes entering in too many significant figures will also result in a lost point. Continuing the above example, if you were to answer 29.4235 cm/s the program will let you slide and give you the point, but if you answer 29.3555 the program will only see 29.3, (that is it does not round up for you) and only looks to the number of sig figs it expects. So again it would be more than 2% out of bounds.

The best bet is to use scientific notation whenever you feel you need to be precise in placing the significant figure, or if the trailing zeros matter!

For all homework from this point forward, the grade assigned by the WebAssign program will be final. So please be cautious in determining significant figures. It matters now as it will later in your career.