TMAT 103

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

ateindetermin is

)0( smeaningles is

)0( 0

0or 0 then 0 If

00

0

0

00

0

0

a

b

baba

a

aa

aa

a

b

§1.2 – Zero and Order of Operations

• Examples – Calculate the followingFind 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 – PEMDAS1. 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

)74(225)27(35

TMAT 103

§1.3

Scientific Notation and Powers of 10

§1.3 – Scientific Notation and Powers of 10

• Powers of 10

• Laws

nn 10 ,10

110

1010

1 and

10

110

10)10(

1010

10

101010

0

nnn

n

mnnm

nmn

m

nmnm

§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 followingWrite 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) Rules1. All non-zero digits are significant2. All zeros between significant digits are significant3. Tagged zeros are significant4. All numbers to the right of a significant digit AND a

decimal point are significant5. Non-tagged zeros to the right in a whole number are

not significant6. Zeros to the left in a measurement less than one are

not significant

§1.4 – Measurement

• Examples – Calculate the followingFind the accuracy (number of significant digits)

of the following:14.7.00000000000814040401404040.00030

§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 followingFind the precision of each of the following:

1.0 m360 V350.000030 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 measurements1. Convert to the same units2. Add or subtract3. Round the result to the same precision as the least

precise of the original measurements

• Multiplying or dividing measurements1. Convert to the same units2. Multiply or divide3. 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 followingFind the sum of: 178m, 33.7m and 100cmFind 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 followingFind the degree of x2yFind 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

1

1

)(

)(

0

a

aa

b

a

b

a

baab

aa

aa

a

aaa

nn

n

nn

nnn

mnnm

nmn

m

nmnm

§1.7 – Exponents and Radicals

• Examples – Simplify the following

3

118

133

26

2

6

26

)(

yx

yx

y

y

y

yy

§1.7 – Exponents and Radicals

• Radicals– Simplifying simple radicals

• Ex:– Simplifying radicals with the following

property:

• Ex:

36

baab

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 followingx2(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 equations1. 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 followingx – 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 followingSolve f = ma for a

Solve e = ƒx + for x

Solve for R3:

321

21

RRR

RRRB

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 followingSolve f = ma for a

when f = 3 and m = 17

Solve e = ƒx + for xwhen e = 11, ƒ = 3.5 and = .01

Solve for R3 when RB, R1, and R2 are all 11

321

21

RRR

RRRB

TMAT 103

§1.13

Applications involving Linear Equations

§1.13 – Applications involving Linear Equations

• Solving application problems1. Read the problem carefully

2. If applicable, draw a picture

3. Use a symbol to label the unknown quantity

4. Write the equation that represents the problem

5. Solve

6. Check

§1.13 – Applications involving Linear Equations

• Examples – Calculate the followingThe 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

• bcadd

c

b

a then If

§1.14 – Ratio and Proportion

• Examples – Calculate the followingFind 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.