Transcript Significant Digits

Physics Introductory Unit
~The Mathematical Background~
Math Skills
• There are several skills, some of which
you have already learned, that you will
need to use extensively in Physics.
• These include the following:
v
d
t
– Algebra (manipulation of formulas)
– Scientific Notation (very lg/sm numbers)
Math
– Significant Digits 0.07034
Rules!
– Unit Conversions 1m  100cm
5.2 104
Scientific Notation
• Scientific notation relies on exponential powers
of ten (10x) to simplify extremely large and small
numbers.
Standard Notation
Scientific Notation
6
4, 673, 000  4.673 10
Power of Ten
Coefficient
• In all cases, numbers written in scientific
notation have a single digit in the ones place
followed by the remaining digits placed to the
right of the decimal point. This is called the
coefficient.
• A multiplied power of ten is indicated afterwards.
Scientific Notation (Cont.)
• Large numbers correspond to positive
powers of ten.
14, 000  1.4 10
4
• Small numbers correspond to negative
powers of ten.
0.00034  3.4 10
4
• Figuring out the power on the ten relates to
how many places you need to move the
decimal point from its initial position.
12080  1.208 10
4 Moves
4
0.037  3.7 10
2 Moves
2
Scientific Notation (Multiplication)
• At times, numbers in scientific notation will be
multiplied as shown below.
6
2
4.2

10
3.1

10



• The trick is to combine the powers of ten with
each other and the non-exponent terms with
each other. Then simplify.
 4.2  3.1 10
13.02  108
1.302  109
6
10
2

Note: Remember
that exponents add
when like bases are
multiplied.
Scientific Notation (Division)
• At times, numbers in scientific notation will be
divided as shown below.
8.4 10 
1.4 10 
3
2
• As before, you need to combine terms. The
exponent rule changes to subtraction when
division is involved.
3

 8.4  10 

 2 
 1.4   10 
6.0  101
Scientific Notation (10x)
• Numbers that are simply powers of ten can be
written in a shorter form without a coefficient.
• Consider the example dealing with 100,000.
100, 000  1.0 105
• In simplified form it can be written as follows:
100, 000  105
• The same holds true for small numbers.
0.001  103
Significant Digits
• Significant digits (sometimes called significant
figures) are those digits that are considered
important in a given number.
• In order to determine which digits are significant,
one must look to the following rules.
– All nonzero digits are significant.
370 or 0.056
– Final zeros after the decimal point are significant.
43.0 or 0.0560
– Zeros between other significant digits are significant.
306 or 0.705
– Zeros used solely for spacing are not significant.
24, 000 or 0.007
Significant Digits (Special Cases)
• A bar can be placed over zeros that are not
normally significant in order to make them
significant.
1 Significant Digit
1 Significant Digit
400 vs. 400
0.003 vs. 0.003
3 Significant Digits
2 Significant Digits
• This usually occurs after some instances of
rounding. Here a problem would specify to how
many digits you must round.
Significant Digits (Rounding)
• Instead of rounding to a place, you round a number
to a specified number of significant digits. This is
done by rounding up or rounding off the number that
would constitute an extra place.
• Round the number 45.63 to 3 significant digits.
– How many significant digits does the number have? 4
– Which digit must be rounded? the 3
Round Off!
– Round up or off? 45.63
45.6
• Round the number 6798 to three significant digits.
6800
Significant Digits (Mult/Div)
• Keeping correct significant digits
while multiplying and dividing relies
on the same process.
– Count the number of significant digits
in each of the numbers being
multiplied or divided.
– Calculate and round your answer to
the number of significant digits found
in the least significant input.
– It is sometimes easier to write these
problems horizontally.
Multiplying
2
3
0.54  6.33
3.4182
3.4
Dividing
4
1
7.261  0.2
36.305
40
Significant Digits (Add/Sub)
• Adding and subtracting rely on the
same process when significant digits
are being kept.
– Align the addends (for addition) or the
minuends and subtrahends (for subtraction)
vertically.
– Draw a vertical line down the least precise
number (the one with least decimal places).
– Add or subtract the values.
– Round to the left of the vertical line.
– Addition problems can have more than two
numbers.
Addition
363.7 14.374
363.7
14.734
378.434
378.4
Subtraction
Units and Unit Conversion
• Anthony jumped in his car and drove 10 to
the grocery store, where he bought 5. He
returned within 30.
WARNING: You will lose points
for any answer that does not
have proper units!!!
Units and Unit Conversion
• In this class we will use the MKS system.
M  meter (m) … unit for length
K  kilogram (kg) … unit for mass
S  second (s) … unit for time
Standard
Units
All other units are derived units … they come
from the 3 above.
Unit Conversions
We can multiply any number by 1 and not
change its value.
1m  100cm
1m
100cm

1
.
100cm 100
cm
How many m are there in 5783cm?
1m
5783cm *
 57.83m
100cm
Practice Problem
6.3hr  ? s
60 min
1
1hr
1hr  60min
.
1min  60sec
60s
1
1min
60 min 60s
6.3hr *
*
 22680 s
1hr
1min
Practice Compound Problem
55
?
mile
hr
1mile  1.61km
1hr  60min
m
s
1km  1000m
1min  60s
.
miles 1.61km 1000m 1hr
1min
55
*
*
*
*
hr 1 miles 1km 60 min 60s
24.6
m
s
Algebra
• Numerous times while studying Physics, you will
be required to use algebra to solve equations.
• Isolating the variable involves the use of inverse
order of operations to manipulate the variables.
– Addition(+) and Subtraction(-) are inverse operations.
– Multiplication(× or ·) and Division(÷) are inverse operations.
– Squaring(2) and square rooting(√) are inverse operations.
Find the value of x
(9  7)*4
x
(4  3) 2 17
When solving for the value of an equation, you must use
ORDER OF OPERATIONS
Parenthesis (Grouping)
Exponents / Powers
Multiplication
Division
Addition
Subtraction
When solving for a variable in an algebraic equation, you
must use
INVERSE ORDER OF OPERATIONS
1) Collect like terms
2) Addition / Subtraction
3) Move variable from denominator to the
numerator
a) Cross multiply
b) Reciprocal
c) Multiply both sides by the variable
4) Multiplication / Division
5) Exponents
6) Parenthesis (Grouping)
Algebra (Sample)
•
•
•
•
Consider the formula shown.
Solve the equation in terms of d.
To do this, we must move t.
What operation is t associated with?
Division
• What is the inverse operation?
Multiplication
• Perform the operation to solve for d.
• Some other problems may involve
more than one step.
d
v
t
d
v t  t
t
d  v t
Other Algebra Samples
• Given the equation:
d
v
t
• Solve for t.
vt  d
d
t
v
• Given the equation:
v22  v12  2a  d2  d1 
• Solve for v2.
v2  
2
v
 1  2a  d2  d1 
Note: When you take the square
root, a  symbol must be included
in front of the radical.
Unit Conversions
Unit Conversions
Unit Conversions
Conclusion
• Physics is a math-based science course.
• All four major skills will come into use
during the course of the year, many as
early as next section.