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TMAT 103
Chapter 1
Fundamental Concepts
TMAT 103
§1.1
The Real Number System
§1.1 – The Real Number System
• Integers
– Positive, Negative, Zero
• Rationals
• Irrationals
• Reals
– Real number line
• Complex Numbers
• Primes
§1.1 – The Real Number System
• Properties of Real Numbers (FYI)
–
–
–
–
–
–
–
–
–
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Distributive Property of Multiplication over Addition
Additive inverse
Multiplicative inverse
Additive identity
Multiplicative identity
§1.1 – The Real Number System
• Signed Numbers
–
–
–
–
–
Absolute Value
Adding 2 signed numbers
Subtracting 2 signed numbers
Multiplying 2 signed numbers
Dividing 2 signed numbers
§1.1 – The Real Number System
• Examples – Calculate the following
|–101|
(- 1½) + (- 2¼)
Bill, a diver, is 120 feet below the surface of the
Pacific Ocean. Heather is directly above Bill in
a balloon that is 260 feet above the Pacific
Ocean. Find the distance between Bill and
Heather.
TMAT 103
§1.2
Zero and Order of Operations
§1.2 – Zero and Order of
Operations
• Operations with 0
a0  a
a0  a
a0  0
If a  b  0 then a  0 or b  0
0
b
 0 (b  0)
a
0
is meaningles s (a  0)
0
0
is indetermin ate
§1.2 – Zero and Order of
Operations
• Examples – Calculate the following
Find values of x that make the following
meaningless:
3x – 7
2x + 1
Find values of x that make the following
indeterminate:
2–x
(2x – 7)(x – 2)
.
§1.2 – Zero and Order of
Operations
•
Order of Operations – PEMDAS
1. Parenthesis
2. Exponents
3. Multiplications and Divisions in the order
they appear left to right
4. Additions and Subtractions in the order they
appear left to right
§1.2 – Zero and Order of
Operations
• Examples – Calculate the following
5  3(7  2)  5  2  2(4  7)
TMAT 103
§1.3
Scientific Notation and Powers of 10
§1.3 – Scientific Notation and
Powers of 10
• Powers of 10
10 n , 10  n
• Laws
10 m 10 n  10 m  n
10 m
mn

10
10 n
(10 m ) n  10 mn
1
1
n
n
10  n and

10
10
10  n
100  1
§1.3 – Scientific Notation and
Powers of 10
• Scientific Notation
– Changing a number from decimal form to
scientific notation
– Changing a number from scientific notation to
decimal form
§1.3 – Scientific Notation and
Powers of 10
• Examples – Calculate the following
Write the following in scientific notation
23700
17070000
.00325
Write the following in decimal form
7.23 x 106
6.2 x 10-3
TMAT 103
§1.4
Measurement
§1.4 – Measurement
• Measurement
– Comparison of a quantity with a standard unit
• In past, units not standard (1 pace, length of ear of
corn, etc.)
– Necessity dictated universally standard units
• Approximate vs. exact
– Accuracy (significant digits)
– Precision
§1.4 – Measurement
•
Accuracy (Significant Digits) Rules
1.
2.
3.
4.
All non-zero digits are significant
All zeros between significant digits are significant
Tagged zeros are significant
All numbers to the right of a significant digit AND a
decimal point are significant
5. Non-tagged zeros to the right in a whole number are
not significant
6. Zeros to the left in a measurement less than one are
not significant
§1.4 – Measurement
• Examples – Calculate the following
Find the accuracy (number of significant digits)
of the following:
14.7
.000000000008
1404040
1404040.00030
§1.4 – Measurement
•
Precision
–
–
–
The smallest unit with which a measurement
is made. In other words, the position of the
rightmost significant digit.
Ex: The precision of 239,000 miles is 1000
miles.
Ex: The precision of 23.55 seconds is .01
seconds
§1.4 – Measurement
• Examples – Calculate the following
Find the precision of each of the following:
1.0 m
360 V
350.000030 V
§1.4 – Measurement
•
Precision and accuracy are different!!!
–
Ex: Determine which of the following
measurements are more precise, and which is
more accurate:
0.00032 feet
23540000 feet
TMAT 103
§1.5
Operations with Measurements
§1.5 – Operations with
Measurements
•
Adding or subtracting measurements
1. Convert to the same units
2. Add or subtract
3. Round the result to the same precision as the least
precise of the original measurements
•
Multiplying or dividing measurements
1. Convert to the same units
2. Multiply or divide
3. Round the result to the same number of significant
digits as the original measurement with the least
significant digits
§1.5 – Operations with
Measurements
• Examples – Calculate the following
Find the sum of: 178m, 33.7m and 100cm
Find the product of: (.065m) and (.9282m)
TMAT 103
§1.6
Algebraic Expressions
§1.6 – Algebraic Expressions
• Terminology
–
–
–
–
–
–
–
Variable
Constant
Term
Numerical coefficient
Monomial, binomial, trinomial, polynomial
Degree of a monomial
Degree of a polynomial
§1.6 – Algebraic Expressions
• Operations on Algebraic expressions
– Adding expressions
– Subtracting expressions
– Evaluating expressions given the values of
variables
§1.6 – Algebraic Expressions
• Examples – Calculate the following
Find the degree of x2y
Find the degree of x2y + w4 + a3b2
(4y + 11) + (11y – 2)
(x2 + x + 17) – (3x – 4)
TMAT 103
§1.7
Exponents and Radicals
§1.7 – Exponents and Radicals
• Laws of Exponents
a m  a n  a mn
am
mn

a
an
(a m ) n  a mn
(ab) n  a n b n
n
an
a
   n
b
b
1
a n  n
a
a0  1
§1.7 – Exponents and Radicals
• Examples – Simplify the following
y6 y2
y6
y2
( y 6 )2
3 13 3
x y 
 8 11 
x y 
§1.7 – Exponents and Radicals
• Radicals
– Simplifying simple radicals
• Ex:
36
– Simplifying radicals with the following
property:
ab  a  b
• Ex:
18
TMAT 103
§1.8
Multiplication of Algebraic
Expressions
§1.8 – Multiplication of
Algebraic Expressions
•
•
•
•
Distributive Property
FOIL
Vertical multiplication
Multiplication of general polynomials
§1.8 – Multiplication of
Algebraic Expressions
• Examples – Calculate the following
x2(y3 + z – 2)
(x + 2)(x – 2)
(3x2 + 4x – 1)(2y – 3z + 7)
TMAT 103
§1.9
Division of Algebraic Expressions
§1.9 – Division of Algebraic
Expressions
• Division by a monomial
• Division by a polynomial
§1.9 – Division of Algebraic
Expressions
• Examples – Calculate the following

14x2 – 10x
2x

6x4 + 4x3 + 2x2 – 11x + 1
(x – 2)

4y3 + 11y – 3
(2y + 1)
TMAT 103
§1.10
Linear Equations
§1.10 – Linear Equations
•
Four properties of equations
1. The same value can be added to both sides
2. The same value can be subtracted from both
sides
3. The same non-zero value can be multiplied on
both sides of the equation
4. The same non-zero value can divided on both
sides of the equation
§1.10 – Linear Equations
• Examples – Calculate the following
x – 4 = 12
4(2y – 3) – (3y + 7) = 6
¼(½x + 8) = ½(x – 16) + 11
TMAT 103
§1.11
Formulas
§1.11 – Formulas
• Formula – equation, usually expressed in
letters, that show the relationship between
quantities
• Solving a formula for a given letter
§1.11 – Formulas
• Examples – Calculate the
following
 Solve f = ma for a
 Solve e = ƒx +  for x
 Solve for R3:
R1 R2
RB 
R1  R2  R3
TMAT 103
§1.12
Substitution of Data into Formulas
§1.12 – Substitution of Data into
Formulas
•
Using a formula to solve a problem where
all but the unknown quantity is given
1. Solve for the unknown
2. Substitute all values with units
3. Solve
§1.12 – Substitution of Data into
Formulas
• Examples – Calculate the following
 Solve f = ma for a
when f = 3 and m = 17
 Solve e = ƒx +  for x
when e = 11, ƒ = 3.5 and  = .01
 Solve for R3 when RB, R1, and R2 are all 11
R1 R2
RB 
R1  R2  R3
TMAT 103
§1.13
Applications involving Linear
Equations
§1.13 – Applications involving
Linear Equations
•
Solving application problems
1.
2.
3.
4.
5.
6.
Read the problem carefully
If applicable, draw a picture
Use a symbol to label the unknown quantity
Write the equation that represents the problem
Solve
Check
§1.13 – Applications involving
Linear Equations
• Examples – Calculate the following
The difference between two numbers is 6.
Their sum is 30. Find the 2 numbers.
The perimeter of an isosceles triangle is 122cm.
Its base is 4cm shorter than one of its equal
sides. Find the lengths of the sides of the
triangle.
TMAT 103
§1.14
Ratio and Proportion
§1.14 – Ratio and Proportion
• Ratio: Quotient of 2
numbers or quantities
• Proportion: Statement
that 2 ratios are equal
• If
a c
 then ad  bc
b d
§1.14 – Ratio and Proportion
• Examples – Calculate the following
Find x:
25 = 75
96
x
The ratio of the length and the width of a
rectangular field is 5:6. Find the dimensions of
the field if its perimeter is 4400m.
TMAT 103
§1.15
Variation
§1.15 – Variation
• Direct Variation
– If 2 quantities, y and x, change and their ratio
remains constant (y/x = k), the quantities vary
directly, or y is directly proportional to x. In
general, this relationship is written in the form
y = kx, where k is the proportionality constant.
– Example:
m varies directly with n; m = 198 when n = 22.
Find m when n = 35.
§1.15 – Variation
• Inverse Variation
– If two quantities, y and x, change and their product
remains constant (yx = k), the quantities vary
inversely, or y is inversely proportional to x. In
general, this relation is written y = k/x, where k is
called the proportionality constant.
– Example:
d varies inversely with e; d = 4/5 when e = 9/16. Find d
when e = 5/3.
§1.15 – Variation
• Joint Variation
– One quantity varies jointly, with 2 or more quantities
when it varies directly with the product of these
quantities. In general, this relation is written y = kxz,
where k is called the proportionality constant.
– Example:
y varies jointly with x and the square of z; y = 150
when x = 3 and z = 5. Find y when x = 12 and z = 8.