1. How scientific measurements should be recorded and used.
Significant Figures
How scientific measurements should be recorded and used.

2. 5.1 cm 2 significant figures

3. How to count significant figures
The number of significant figures is determined by counting the first nonzero digit and all following digits in a number with a decimal.
1.34 has 3 sig figs
In the United States use a period (.) not a comma (,) for a decimal point.

4. Estimate 5.11 cm for 3 sig figs

5. Internal zeroes
Internal zeros (zeros between nonzero digits) are always significant.
has 5 sig figs.
has 4 sig figs.

6. A little shorter could be 5
A little shorter could be 5.09 cm The internal zero would be significant

7. How could we get another sig fig?
We might get a better scale with scale markings between 5.1 and 5.2 so there are 10 divisions 5.11, 5.12, 5.13, etc. and pick the closest line again and estimate a 4th sig fig.

8. What if we changed the units?
5.11 cm = ? Km
5.11 cm = 5.11 cm
5.11 cm = km
How many sig figs is this?

9. Initial zeroes
Zeros before the first nonzero digit are never significant km has 3 sig figs The measurement was 3 sig figs. Changing the units does not make the measurement better.

10. Final zeroes Final zeros in a decimal number are always significant.
4.500 has 4 sig figs.
has 3 sig figs

11. Final zeroes when NO decimal
Numbers without decimals may be:
Exact: 1 mile = 5280 feet (by definition)
Countable: 1 ream = 500 sheets
Approximate: The distance to the sun is ,000,000 miles
Exact and countable numbers should be written without decimals.

12. Exponential Notation Scientific Notation
93,000,000 miles could be written as 93.x106 miles if it is meant to convey 2 sig figs
Scientific Notation
93,000,000 miles could be written as
9.3x107 miles if it is meant to convey 2 sig figs or
9.30x107 miles if it is meant to convey 3 sig figs

13. Calculate the area of a rectangle 5. cm x 80.2 cm
The area is cm
x 5. cm
401.0 cm²
Is how many significant figures?

14. In multiplication and division
In multiplication and division the calculated answer should have no more significant figures than the measurement with the least significant figures.
0.044 =  5.9 (2 sig figs)
134.56/0.044 =  3.1x103
not
5. cm x 80.2 cm = cm² → 4.x10² cm²

15. Calculate the area of a rectangle 5. cm x 80.2 cm
What we really want to know is how many boxes
there are 1 cm x 1 cm

16. Calculate the area of a rectangle 5. cm x 80.2 cm
To be 5. cm to 1 sig fig means it is between 4.5 and 5.5
To be 80.2 cm long to 3 sig figs means it is between
80.15 and cm long.

17. Calculate the area of a rectangle 5. cm x 80.2 cm
By error analysis the maximum possible area is:
5.5 cm x cm = cm²
4.5 cm x cm = cm² is the minimum
Or ±

18. Calculate the area of a rectangle 5. cm x 80.2 cm
Area is ± cm²
Because of the 1 sig fig width the number of cm² boxes
could be as much as 40 more or 40 less than 401 cm².
That would leave the answer good to only 1 sig fig.

19. Calculate the length of the perimeter of the rectangle
5. cm
80.2 cm
This would be cm
5. cm
+80.2 cm
170.4 cm How many sig figs?

20. In addition and subtraction
In addition and subtraction the calculated answer should carry as many decimal places (not significant figures) as the measurement with the fewest decimal places.
+ 2.36 →
71.2 →

21. Calculate the length of the perimeter of the rectangle
5. cm
80.2 cm
This would be cm
5. cm
+80.2 cm
170.4 cm → cm

22. Again using error analysis, the perimeter is:
max min or ± 1.1 cm
So it is surely not good to the nearest 0.1 cm.
But it is nearly good to 3 sig figs.
Significant figures approximate a more careful error
analysis without nearly as much effort.

23. Round Numbers Correctly
Round the answer to the nearest number having the correct number of significant figures.
If the leftmost digit being removed is less than 5, drop this digit and all following digits.
If the leftmost digit being removed is 5 or greater, drop this digit and all following digits AND increase that rightmost digit being retained by one unit.

24. Rounding Example 457.26451 is 457.2645 to 7 sig figs
4.6x102 to 2 sig figs

25. Round at the end of a calculation
Calculate an answer to a problem using all given significant figures, then round the final answer to the correct significant figures.
Consider each step along the way to determine correct significant figures, but do not round at each step along the way.
(4.65 ─ 2.2)×3.51 = → 8.6