Significan Digits

1
Significan Digits

• aka
• Significant Figures

2
What are they?

• They show the certainty (precision) of your
  measuring device.
• 0.1g 0.1000g
• Whats the difference in these two numbers?
• Sig. Figs. are only for measurements. So always
  ask, is the value you are looking at a
  measurement of some kind. The sig. figs. for
  exact values are ignored.
• The last digit is always the digit that is least
  certain (the digit you are estimating).

3
More Difficult Rules

• (1) All nonzero digits are significant
• 1.234 g has 4 significant figures, 1.2 g has 2
  significant figures.
• (2) Zeroes between nonzero digits are
  significant
• 1002 kg has 4 significant figures, 3.07 mL has
  3 significant figures.
• (3) Leading zeros to the left of the first
  nonzero digits are not significant such zeroes
  merely indicate the position of the decimal
  point
• 0.001 oC has only 1 significant figure, 0.012 g
  has 2 significant figures.
• (4) Trailing zeroes that are also to the right of
  a decimal point in a number are significant
• 0.0230 mL has 3 significant figures, 0.20 g has
  2 significant figures.
• (5) When a number ends in zeroes that are not to
  the right of a decimal point, the zeroes are not
  necessarily significant
• 190 miles may be 2 or 3 significant
  figures, 50,600 calories may be 3, 4, or 5
  significant figures.
• The potential ambiguity in the last rule can be
  avoided by the use of standard exponential, or
  "scientific," notation. For example, depending on
  whether the number of significant figures is 3,
  4, or 5, we would write 50,600 calories as
• 5.06 104 calories (3 significant
  figures) 5.060 104 calories (4 significant
  figures), or 5.0600 104 calories (5
  significant figures).
• Another potential ambiguity comes from an ending
  decimal
• 50,600. calories (5 significant figures)
• The decimal makes all digits significant

4
How do you find how many sig. figs. a number has?
If a decimal point is Present, ignore zeros on
the Pacific (left) side (except if there are
zeros between non-zero numbers in this case all
are significant from the first non-zero number
on). If the decimal point is Absent, ignore zeros
on the Atlantic (right) side. Everything else is
significant.
Ex 1) 0.003200 (4 sig. figs) 2) 1.003200 (7
sig. figs)
5
Doing math with Sig. figs.!

• Adding and subtracting line the numbers
  decimal points up. Add (or subtract) the
  numbers.
• To decide where to round, the result should have
  as many decimal places as the measured number
  with the smallest number of decimal places.
• Ex 121.23 (two decimal places) 1.2540 (four
  decimal places) 122.4840
• Answer should round based on two decimal places
  which would give 122.48

6
Example 1

• A) 13.559 1.22599 20.23
• B) 100.9552 29.059 ? 79.8
• C) 0.98 0.099 0.422

7
Rounding to the right number of sig. figs.

• Given a value, start from the left (using the
  concept of pacific ocean for decimal numbers) and
  begin counting the number of sig. figs. you are
  told to have.
• Replace whole numbers with zeros for place
  holders, decimal numbers just leave them off.

8
Example 2

• Round each number to three significant digits
• 0.0050505
• 123040
• 3500
• 2.0309
• 1.4592

9
How to multiply and divide?

• Multiply or divide the numbers given.
• Count the number of sig.figs. in each value.
• Round your answer to the number of sig. figs.
  that has the least of the values.
• So if you are multiplying or dividing two values
  one with 3 sig. figs. and one with 2 sig. figs.,
  your answer should only have two sig. figs.

10
Example 3

• 25 x 2.39 x 0.1
• 950/0.0359
• 7.228 x 40.3
• 3.4
• D) 6.53 x 0.00042

11
Add/subtract with multiply/divide

• Break into components (like operations) and
  perform an order of operations. Follow individual
  rules for each component.
• PEMDAS (Parenthesis, Exponent, Multiplication,
  Division, Addition, Subtraction)
• 1. Parenthesis
• 2. Multiply/Division
• 3. Addition/Subtraction
• Ex 3.2 2.35 x 1.560 vs. (3.2 2.35) x 1.560
• 6.9 8.7

12
Example 4

• (0.0035 0.0021)
• 0.0245
• 375 27.33 x 25
• 4.523
• (2.2 2.1)
• D) 0.036 3.2/1.37

13
Scientific Notation

• Its the way to handle very large or small
  numbers.
• Ex 0.0000000000032 is a very small number
• Ex 1,231,500,000 is a very large number
• To go from a number to scientific notation
• 1) Find the first non-zero value and place a
  decimal in between it and the next number
  (located to the right).
• 2) Now, starting from your newly added decimal,
  count how many spaces are between the decimal you
  added and the original decimal. Each space
  represents a factor of 10 (why you see x 10x). If
  there is no decimal count from your newly added
  decimal to the end of number.
• 3) If you move left, your exponent (superscript)
  x will be negative the number of spaces counted.
  If you move right, your exponent (superscript) x
  will be positive the number of spaces counted.
• 4) Keep number of sig. figs. In original value.

14
Scientific Notation

• Its the way to handle very large or small
  numbers.
• Ex 0.0000000000032 is a very small number
• 1) 0.000000000003.2
• 2) There are 12 spaces to the left
• 3) Left tells you x 10-12 and negative
• 4) There were two sig. figs. to begin with so
  converted value is now
• 3.2 x 10-12
• Ex 1,231,500,000 is a very large number
• 1) 1.231500000
• 2) There are 9 spaces to the right (end of
  number)
• 3) Right tells you x 109 and positive (we do not
  put )
• 4) There were 5 sig. figs. to begin with so
  converted value is now
• 1.2315 x 109
• You can also work backwards from scientific
  notation to number. If the exponent (superscript)
  is positive move that many spaces right and fill
  in zeros. If exponent (superscript) is negative,
  move that many spaces left and fill in zeros
  (keep decimal).

15
Example 5

• 0.00068 (give scientific notation)
• 1.38 x 105 (give number)
• 585,000 (give scientific notation)
• D) 6 x 10-6 (give number)
• E) 695.1 x 10-4 (give standard scientific
  notation tricky)

16
Example 6

• Sig. Figs with scientific notation. How many sig.
  figs. does each number have?
• 6.80 x 10-3
• 1.3821 x 105
• 3 x 108
• Round each to two sig. figs.

17
Sig. Figs. and Actual Measurements

• The number you estimate determines the sig. figs.
  
• You can tell this is between 2.8 and 2.9. What
  numbers come between 2.8 and 2.9?
• 2.81, 2.82, 2.83, etc
• 2.811 comes between 2.8 and 2.9 as well but in
  this case you would be estimating the last two
  decimal places.
• -In this case the hundredths place determines the
  sig. figs. because it is the number you are
  estimating.
• - Rule of thumb when taking a measurement, the
  number of sig. figs. will be a tenths place to
  the right of the decimal place you can actually
  read with a tick mark.

18
Practice