Session 1 - Math Concepts Review

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Principles of Physics &
Instrumentation of Medical
Sonography
Math Concepts Review
Math Concepts Review
Reciprocals
Exponents
Significant Figures
Exponential Notation
Units
Unit Prefixes
Unit Conversion
Rules of Algebra
Relationships
Reciprocals
The reciprocal or inverse of a number is ‘1’ divided
by that number. Any number multiplied by its
reciprocal equals one.
The reciprocal of 250 is 1/250 or 0.004
250 x 1/250 = 1
What is the reciprocal of:
3?
111 ?
.5 ?
Exponents
When we write 23, we mean
2 multiplied by itself 3 times
2 x 2 x 2 = 8.
or
The numeral 2 is the base
The superscript numeral 3 is the exponent
When written as an unknown variable,
an is ‘a times a taken n times’.
Two special cases are:
a1=a and a0=1
Exponents - Multiplication
Multiplying two numbers
expressed as powers of the
same base (an)(am) = an+m
Exponents - Division
Dividing one number raised to a power
by another number of the same base -
an = an-m
am
Knowing: 1 = a0
Then: 1 = a0
am am
So… 1 = a-m
am
=
a 0-m
Exponents
When the exponent is a fraction
(a1/n), it is called a root.
If a = 8 and n = 3, then…
a1/n = 2 (the cube root of 8)
Significant Figures
A method of keeping track of numbers is to
write only those figures that are significant.
The zeros in 0.053 and 5,300 are not
significant when using scientific notation,
while the zero in 1.053 is.
When zeros are “place keepers” for the
decimal point, as in 5,300 and 0.053, they
are not significant.
Exponential Notation
Writing out very small or large
numbers is inefficient; so
exponential notation is used.
There are 3 separate types of numbers
involved in exponential notation:
1. The first number in the notation is a
whole number between and including
1 - 9, followed by a decimal point
and any remaining significant numbers
of the original number.
2. The second number represents the
base which is 10.
Expotential Notation
3. The final number is the exponent. The exponent
is the “power” to which 10 is being raised. Its
value can be positive or negative.



If the exponent is positive,
the resulting number > 0
If the exponent is negative, the
resulting number < 0
If the exponent is 0, the resulting
number will be the same as the first
number of the equation
Example I
Express 50,000 in exponential notation:
1. If the number  1, then place the
decimal point 1 place after the first
number farthest to the left. If the
number < ‘1’, then place the decimal
point 1 place after the first non-zero
number going from left to right.
5.0000
Express 50,000 in exponential notation
2. Place the multiplication sign (x) after
that number.
5.0000 x
3. Count the number of places from
the last number to the decimal point
& note your direction (in this case
there are 4 places to the left).
Express 50,000 in exponential notation
4. The number of places will be the
exponent; a left direction makes the
exponent positive, whereas a right
direction will make the exponent a
negative.
5.0000 x 104
Since we are trying to be efficient,
the 4 non-significant zeroes can be
removed:
5 x 104
Example II
Express 0.0001074 in
exponential notation:
1. If the number  1, then place the
decimal point 1 place after the first
number farthest to the left. If the
number < 1, then place the decimal
point 1 place after the first non-zero
number going from left to right.
1.074
2. Place the multiplication sign (x)
after that number.
1.074
x
3. Count the number of places from
the last significant number to the
decimal point and note your
direction (in this example there
are 4 places to the left)
4. The number of places will be the
exponent; a left direction makes the
exponent positive, whereas a right
direction will make the exponent
negative.
1.074
x
10-4
This is as efficient as we can be;
the zero is significant & cannot be
removed
Practice: Express these numbers
in exponential notation
a) 0.086010
b) 90,270,000,000,000,000
c) 0.0000000544
d) 360,000,000,000,001
Exponential Notation
When performing addition or
subtraction on exponential notation,
both numbers must be expressed in
the same power of 10. If they are
not, convert the number(s) so that
they are raised to the same power.
Examples
Addition:
(a x 10n) + (b x 10n) = (a + b) x 10n
Subtraction:
(a x 10n) - (b x 10n) = (a - b) x 10n
Examples
Multiplication:
(a x 10n) x (b x 10m) = (ab) x 10n+m
Division:
a x 10n
b x 10m
=a
n-m
X
10
b
UNITS
A unit is a predetermined quantity used as a
standard of measurement. The unit tag
gives a meaning to the number, it tells us
what is being measured.
Examples – inch, mile, gallon, square foot
However, the medical and scientific world
uses the Metric System S.I. (Systeme
Internationale-French)
Examples – centimeter, kilometer & liter
Measurement units we will be using:
Unit of Length: meter (m)
Unit of Mass: gram (g)
Unit of Volume: liter (l)
Unit of Time: second (s)
Each of the units of measurement
can be made smaller or larger
by adding a unit prefix
Formulas we will be using:
Area: Consists of two dimensions
length (m) x width (m) = (m2)
Volume: Consists of three dimensions

Volume of a rectangular prism in cm:
length (L) x width (W) x height (H) = cm3

Volume of a cylinder:
V =  r2 H
Unit Prefixes
Factor
Prefix
Powers
of Ten
Meaning
Symbol
giga
1,000,000,000
109
billion
G
mega
1,000,000
106
million
M
kilo
1,000
103
thousand
k
hecto
100
102
hundred
h
deca
10
101
ten
da
deci
1/10 (0.1)
10-1
tenth
d
centi
1/100 (0.01)
10-2
hundredth
c
milli
1/1,000 (0.001)
10-3
thousandth
m
micro
1/1,000,000
or (0.000001)
1/1,000,000,000
or (0.000000001)
10-6
millionth

10-9
billionth
n
nano
nano
micro
milli
centi
deci UNIT
Deca
Hecto
Kilo
Mega
Giga
Unit Conversion
To compare 2 numbers or to perform
calculations, we must find the factor
required to get the corresponding new unit.
1. Write out what you know from the math
question.
2. Find the factor (in a table or memorize)
3. Multiply the beginning number by the factor
and change the answer to the new unit tag
micro
10-6
milli
10-3
centi
deci
Meter
10-2
10-1
100
Convert 25.4 millimeters to centimeters
1. Write what you start with: 25.4 mm.
2. Centi is to the right of the milli, we will change
to larger units & get a smaller number
3. Determine the factor (10-3/10-2 = 10-1 = .1)
4. Multiply the beginning number by the factor
& change the answer to the new unit tag:
25.4 mm X .1 = 2.54 cm
micro
10-6
milli
10-3
centi
deci
10-2
10-1
Second
100
Convert .4 seconds into microseconds
1. Write what you start with: .4 s
2. Microsecond is to the left of second, we will
change to smaller units & get a larger number
3. Determine the factor:100/10-6=106 (1,000,000)
4. Multiply the beginning number by the factor and
change the answer to the new unit tag
4 s X 1,000,000 = 400,000 s
Practice: Convert 3.5 Megahertz (MHz) to Kilohertz (KHz)
Rules of Algebra
Solving for an unknown
variable with known variables
Whatever you do to one side of the
equation, you must do on the other side.
To solve for an unknown variable, it must be
by itself, so we have to move all other
variables to the other side of the equal sign.
Recall: any symbol can be substituted for
the letters in this equation
Addition
Solve for X, Y=15 & Z=30: X + Y = Z
X + Y (-Y) = Z (- Y)
X=Z–Y
X = 30 – 15
X = 15
Subtraction
Solve for X, Y=15 & Z=30: X – Y = Z
X – Y (+Y) = Z (+ Y)
X=Z+Y
X = 30 + 15
X = 45
Division
Solve for X, Z=12 & Y=3:
XY= Z
XY (Y) = Z (Y)
X = ZY
X = 12  3
X=4
Multiplication
Solve for X, Z=12 & Y=3: X/Y = Z
X/Y (x Y) = Z (x Y)
X=ZxY
X = 12 x 3
X = 36
Relationships:
Unrelated: when you change one thing;
the other factor does not change.
Inversely Related: When you increase or
decrease a variable; the other variable
changes in the opposite direction
Directly Related: When you increase or
decrease a variable; the other variable
changes in the same direction
Inversely Related
B
A=C
In the above equation, variables A & C are
inversely related. This means that when the
value of A is changed; the value of C must
change in the opposite direction (and vice
versa).
To be inversely proportional the variables of
interest must be:
on opposite sides of the equal sign and
on opposite sides of the divisor line.
Directly Related
A=B
C
In the above equation, variables A and B are
directly related. This means when the value
of A is changed; the value of B must change
in the same direction (and vice versa).
To be directly proportional the variables of
interest must be:
on opposite sides of the equal sign and
on the same side of the divisor line.
Direct Linear Proportionality
Whenr A and B are directly related and
whenever the value of A changes (increased or
decreased), the value of B changes in the same
direction by the same percentage.
In a direct linear proportion, the variables of
interest must be:
on opposite sides of the equal sign and
on the same side of the divisor line and
when A is doubled, B doubles.
If A is cut in half, B is also cut in half
(or in the same percentage proportion)
Direct Linear Proportionality
Y = 3X; Substitute 1, 2 & 4 for X
Y is direct linear proportional to X
When you graph the 2 dependent variables
(Y & X); the result is a straight line.
Y
o
12
10
8
o
6
4
2
o
1
2
3
4
5
X
Direct Non-Linear Proportionality
Whenever A and B are directly related and
whenever the value of A changes (increased or
decreased), the value of B changes in the same
direction but NOT by the same percentage.
In a direct non-linear proportion, the variables
of interest must be:
on opposite sides of the equal sign and
on the same side of the divisor line and
when A changes, B changes in the same
direction but at a different percentage.
Direct Non-Linear Proportionality
Y = 3X2; Substitute 1, 2 & 4 for X
Y is direct linear proportional to X
When you graph the 2 dependent variables
(Y & X); the result is NOT a straight line.
Y
60
o
50
40
30
20
10
o
1
o
2
3
4
5
X
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