Significant Figures PowerPoint

Significant Figures
Mathematics Involving Measurements Are
Probably Different Than What You Are
Used To.

Exact vs. Measured Values
Most of the math we are taught involves using
exact numbers.
Exact numbers are used in science along with
measurements, which are inexact numbers.
Counted objects or defined values are exact.
For example, the average person has 10 fingers.
That’s a count.
There are exactly 1000 mm in 1 m. That’s a defined
value.

Significant Figures
So what the heck are significant figures, and
how do we use them?
Any measured value is a significant figure.
If you measure a toothpick to be 4.72 cm long,
there are 3 significant figures in the value: 4,7
and 2.
If the toothpick measured 4.70 cm, there are still
3 significant figures.

Significant Figures
This brings up an interesting question: Are there
digits that are not significant?
The answer is, Yes!
All non-zero digits are automatically significant.
And zeros that show precision are significant.
However, zeros that only serve as place holders,
that is, they locate the decimal in a number, are
not significant!

Significant Figures
A good guideline to follow is this: If zeros appear
or disappear when you change the units of the
measurement, the zeros are not significant.
For example, 5 m has 1 significant figure.
But 5 m = 5000 mm. Are the 3 zeros significant?
No! They simply appeared because we changed
units. Both numbers have 1 sig. fig.

Significant Figures
So, zeros at the beginning of a number (like
0.00065) are always place holders and are never
sig figs.
And, zeros at the end of a number where the
decimal place is implied (like 30,000), are also
place holders and are not sig figs.

Significant Figures
But, zeros in the middle of a number (like 605)
are not place holders. They are part of the
measurement and are always sig figs.
And, most importantly, zeros at the end of a
number with a decimal place show precision.
They are not place holders and are sig figs.

Significant Figures
This is one of the things your math teachers
probably didn’t tell you.
2.2 g ≠ 2.200 g
The second measurement is much more precise
than the first! The 2 zeros at the end of 2.200 g
show precision and are sig figs.
2.2 g has 2 sig figs, but 2.200 g. has 4!

Significant Figures
So the trick is to know when zeros are significant
and when they are not.
600 has only 1 sig fig, but 60.0 has 3.
But, what happens if you were measuring the
length of a table to the nearest 1 cm and it came
out to be exactly 300 cm long?

Significant Figures
If the table had been 299 cm, there would be 3
sig figs. Likewise if it had been 301 cm, 3 sig
figs, right?
How then do we show that the last zero in 300
cm is significant?
The answer is to put a bar over (or under) the
zero to show that it was a measured value.

Significant Figures
To show that we measured our table to exactly
300 cm, we write it this way: 300 cm.
Since zeros at the beginning of a number with a
decimal place are never significant, and zeros at
the end of a number with a decimal are always
significant, the only time we need a zero with a
bar is in a number with no decimal place.

Significant Zeros
A simple way to think about zeros is this:
If zeros appear or disappear when you change
the units or put a number in scientific notation,
they were not significant figures.
Ex: 6 m has 1 sig fig. 6 m = 6000 mm: still 1 sig
fig.
0.005 s has 1 sig fig. 5 x 10 -3 s has 1 sig fig.

Scientific Notation
A word about scientific notation, numbers in
exponential form.
We use them frequently in chemistry, because
we often have to deal with very large or very
small numbers. And all those zeros get to be a
pain.

Scientific Notation
The classic example involves a number that
chemists use all the time. The number is
602,000,000,000,000,000,000,000
It would be ridiculous to write the number out this
way each time, so instead we use scientific
notation: 6.02 x 10 23 . Much more efficient.
This number has 3 sig. figs: 6, 0 and 2 in both
the exponential and non-exponential forms.

Time to try your hand at sig figs.
How many sig figs in
each of these values?
3.041 m
0.000050 m 2
3,480,200 yr
-1.200°C
1.0800 x 10 -2 L
6000 m
Let’s see how you did.
Here are the answers
4
2
5
4
5
3

Still Confused?
It is not unusual for people to still be a little hazy
on significant figures at this point.
So here’s a trick to help you figure out which
digits are significant and which are not.
It is called the Atlantic/Pacific model.
Your teacher will show you the model on the
board.

Rounding
Round each to 4 sig figs.
12,345,670
2.35500
1.4638 x 10 2
0.000657030
100,250.1
23,475
23,485
Here are the answers:
12,350,000 (1.235 x 10 7 )
2.355
1.464 x 10 2
0.0006570 (6.570 x 10 -4 )
100,300
23,480
23,480

Wait a Minute!
No, that last answer was not a typo.
This is another thing your math teacher probably
didn’t tell you.
When you learned to round, in grade school,
your brain wasn’t ready for this idea.
But now you are!!! (Isn’t that exciting?)

Rounding Exact Fives
Five is that magic number that we are not sure
about when rounding. To make things easy
when you were younger, your teachers said to
always round up.
Scientists round exact 5’s so that the rounded
digit to the left of the 5 becomes an even
number.

Rounding Exact Fives
This eliminates the mathematical bias introduced
by always rounding 5’s up.
So, rounding to the nearest 10, the number 75
becomes 80 (just like you were taught), but 85
also rounds to 80. Eight is even, where the 9 of
90 is odd.
5.5 becomes 6, and 4.5 becomes 4.
What about 4.51?
This is not an exact 5. It’s more than 5, so round
up: 5.0