Math110F04

Notes

 

USE THE REAL LINE TO ILLUSTRATE THE NUMBERS

- draw the line on the board and then "ornament it" as you explain each kind of number.

 

1.1 REAL NUMBERS ¾  numbers some in various kinds

- POSITIVE INTEGERS the counting number 1, 2, 3, 4 … are called NATURAL NUMBERS or WHOLE NUMBERS

-  Zero was a discovery, the number that stands for NOTHING and is considered one of the INTEGERS

- INTEGERS are …, -4, -3, -2, -1, 0, 1, 2, 3, 4, … where the … means "and so it continues" so the INTEGERS are the WHOLE NUMBERS, ZERO and the NEGATIVE WHOLE NUMBERS.

SYMBOLS ¾ like a, b, c, d … x,y, … are commonly used to represent ARBITRARY REAL NUMBERS (We will see that Real Numbers include much more than just Integers)

OPERATIONS ¾ the simplest operation between two numbers is an assertion of equality of inequality. EQUAL '=' is shown as a = b, meaning that the number that a stands for is the same number that b stands for.  INEQUALITY '¹' a ¹ b means that a and b stand for DIFFERENT numbers.  An EQUATION is a statement that says that one set of symbols on the left side of an equals sign has the same value as another set of symbols on the other side of an equals sign.

If c = ab, this means that c has the same value as a MULTIPLIED by b.  We can also write this as a×b with a little dot indication multiplication.  Or we can set off the elements in PARENTHESIES as c = (a)(b).  The two numbers a and b are call FACTORS of c, or DIVISORS of c since either one will divide c since c is the PRODUCT of the two Factors.

PRIME NUMBERS ¾ a positive number other than 1, p, is PRIME if its factors are only 1 and p.

ex. 2, 3, 5, 7, 11, 13, 17, 19 …

FUNDAMENTAL THEOREM OF ARITHMETIC — Every positive integer different from 1 can be expressed as a product of primes in one and only one way (except for the order of the factors).  ex. 12 = 2×2×3, 126=2×3×3×7

KINDS OF NUMBERS --

            WHOLE NUMBERS — non-negative integers

            INTEGERS—positive & negative whole numbers & 0: …-4,-3,-2,-1,0,1,2,3,4, …

            PRIME — only have 1 and themselves as FACTORS

            RATIONAL — a real number than can be expressed as a ratio of integers a/b, b¹0

            IRRATIONAL — numbers that are not rational, ex. p»3.14159… or

            REAL — all the Rational and Irrational number taken together.

REAL NUMBERS ARE CLOSED RELATIVE TO ADDITION — What this means is that any time you add to real numbers together you get another real number.

REAL NUMBERS ARE CLOSED RELATIVE TO MULTIPLICATION — mean any time you multiply real numbers you get another real number.

 

 

 

 

PROPERTIES OF REAL NUMBERS (see table on page (4))

ADDITION

- is COMMUTATIVE => a+b=b+a, Order of operation is immaterial when adding

- is ASSOCIATIVE => (a+b)+c = a + (b+c), Grouping is immaterial to the result

ADDITIVE IDENTITY — a + 0 = a, adding zero doesn't change the result

ADDITIVE INVERSE — a + (-a) = 0, the negative of a number added to the number is 0

 

MULTIPLICATION

- is COMMUTATIVE (ab = ba) => Order of operation is immaterial.

- is ASSOCIATIVE ( a(bc)=(ab)c => Grouping is immaterial to the result

MULTIPLICATIVE IDENTITY — a×1=a, Multiplying by one leave a unchanged

MULTIPLICATIVE INVERSE — for a¹0 is 1/a the reciprocal, a×(1/a) = 1.

DISTRIBUTIVE — Multiplication is DISTRIBUTIVE over Addition, means:

- a(b+c)=ab + bc  and (a+b)c=ac+bc, i.e. it doesn't matter which operation you do first, you can multiply through or you can add and then multiply.

Ex. (p+q)(r+s)=pr+ps+qr+qs.

PROPERTIES OF EQUALITY — if a=b and c are real numbers, then you can SUBSTITUTE a for b or b for a so a+c=b+c, and ac=bc. (5)

PRODUCT INVOLVING ZERO — is 0 is a factor in any product the result is zero.  So if the result of a product is zero, one or more of the factors must be zero.  This is called the ZERO FACTOR THEOREM.

NEGATIVE NUMBERS — basically the product of two number is: 1) POSITIVE if both numbers have the same sign, ex. -1×-4=+4, or 2×4=+8, -6×-2=+12, and 2) NEGATIVE if they have different signs, ex. -2×+4=-8, _6×-3=-18

NOTATION FOR RECIPROCALS — if a¹0 then 1/a can also be written a^-1 i.e. with a superscript (exponent) of minus one.  a×a^-1=a×1/a=1

SUBTRACTION — is the same as ADDING THE NEGATIVE NUMBER,

- so a – b = a + (-b)

DIVIDING — is the same as multiplying by the reciprocal so a/b = a×1/b=a×b^-1 b¹0

- a/b or  are referred to as QUOTIENT of a and b or the FRACTION a over b where a is the NUMERATOR (the number on top) and b is the DENOMINATOR the number on the bottom.  Since we cannot divide by zero the fraction a/b is NOT DEFINED if b = 0.  That's why we have always included restrictions like b¹0.

THE REAL LINE — real numbers can be Represented by a line l such that each real number is a POINT on the line.  Each point corresponds to one real number.  This is called ONE TO ONE correspondence.  The point zero (0) is called the ORIGIN.  The number a that corresponds to a particular point is called the COORDINATE of A.  A set of coordinates is referred to as a COORDINATE SYSTEM.

ABSOLUTE VALUE — sometimes we want to ensure that a number is positive.  When that is the case we take its absolute value, written | a | where two vertical lines around the symbol indicate its absolute value.  If a is negative then taking the absolute value will make it positive.  If it is already positive, then it won't change it.

DISTANCE BETWEEN COORDINATES — the distance between two coordinates on the REAL LINE d(A,B) = | b – a |  — illustrate on the BB.

SCIENTIFIC NOTATION — a = c´10ⁿ where 1£c<10 and n is an integer

- ex. 1.24´10² which can also be written 124.  It is particularly useful for writing very large or very small numbers in a compact form, ex. 5.3´10^-23 grams which is the approx. weight of an oxygen molecule 0.000 000 000 000 000 000 000 053 grams

- SIGNIFICANT FIGURES when we write in scientific notation, but also in ordinary notation the idea of accuracy or significance is important.  We can only expect an answer to be accurate to the level we measure, ex. the nearest millimeter will express meters to one part in a thousand.  The nearest inch will measure feet to one part in twelve. If an exact answer is say 37.2638 and we want to express it to FOUR SIGNIFICANT FIGURES we say it is 37.26 and lop off the .0038.  Expressing it in Scientific Notation would be 3.726´10¹

 

EXERCISES: There are 63 problems on pages 16-18.  Most are very easy.  Do a selection from each Exercise Group.  Notice that the answers to the odd problems are given in the back of the book.  A good strategy is to do the odd number problems without consulting the book and then check your work against the answers in the back of the book.  If your answer is wrong then you know you missed something.  Study that and correct your answer.

Section 1.2 (19) Exponents and Radicals

aⁿ means a×a×a … n-times. ¾ n is called the EXPONENT and the real number, a, to which it is applied is called the BASE ¾ we talk about this term as "a to the nth power" ¾ do some examples

 

caⁿ -- c here is called the COEFFICIENT so caⁿ means (c)aⁿ not (ca)ⁿ you can use parentheses to make it clear what term the EXPONENT applies to.  (18)

 

MULTIPLICATION OF TERMS WITH THE SAME BASE AND DIFFERENT EXPONENTS ¾ aⁿ×a^m = (a×a×a×…×a)×(a×a×a×…×a) = a^n+m

 

a⁰ = 1, a BASE raised the zero power = if a⁰×a¹=a⁰⁺¹=a¹ so we can divide through and show that a⁰=1

 

(a^m)ⁿ = a^m×a^m×a^m× … n-times so (a^m)ⁿ = a^mn when you raise a BASE that is raised to a power by another power the result is the PRODUCT of the powers.

 

WHAT ABOUT NEGATIVE COEFFICIENTS?

 

a×(1/a)=1 which is a¹×a^? = a^1+? = a⁰ = 1 ¾ so ? must be –1 so (1/a) = a^-1

and a^-n = (1/a)×(1/a)× … n-times.

SO TAKING THE RECIPROCAL CHANGES THE SIGN OF THE EXPONENT!

Some Examples:

  etc. see page 22

 

ROOTS 1.  if a = 0 then =0

IF 2. a>0 then  is a POSITIVE REAL NUMBER b such that bⁿ=a.

    3a. IF a < 0 and n is ODD then  is the NEGATIVE real number b such that bⁿ=a

    3b. IF a < 0 and n is EVEN then  is NOT A REAL NUMBER

IF n = 2 then  can be written as  and call it the PRINCIPLE SQUARE ROOT of a or simply the SQUARE ROOT of a. 

 

See tables on page 24.  GO OVER THE PROPERTIES

 

REMOVING nth powers (25)

Removing FACTORS from a RADICAL – find the largest factor that can be expressed with the same exponent as the radical and take it out.  Ex. sqrt(8)=sqrt(4x2)=2sqrt(2)

 

Rationalizing Denominators of Quotients – the general strategy is to MULTIPLY BY ONE

-- find the radical term which multiplied by the denominator generates a whole term and multiply top and bottom by that term..  SEE TABLE on page 26.

 

DEFINITION OF RATIONAL EXPONENTS – (27) see green box on page 27

 

Do some example problems.

 

(31) SECTION 1.3 Algebraic Expressions

SETS – a set is a collection of objects.  We denote Sets with capital letters and a particular member or element of a set with the corresponding lower case letter. SET THEORY is used to talk about mathematical relationships.

 

EQUALITY – two sets are equal if they have the same elements.

NOTATION = equality ¹ inequality Î set membership (in the set) Ï not in the set.

 

VARIABLES and CONSTANTS

Constant – a letter or symbol that represents a SPECIFIC ELEMENT of a set., ex. 5, pi etc.

Variable – a letter or symbol that represents an arbitrary element ANY element of the set, say x denoting a REAL NUMBER.

CONVENTION – letters near the end of the alphabet usually represent VARIABLES and letters near the beginning of the alphabet usually represent CONSTANTS.

 

ALGEBRAIC EXPRESSION (32) beginning with a Collection of CONSTANTS and VARIABLES an ALGEBRAIC EXPRESSION is one which is the result of applying

1) Additions, 2) Subtractions, 3) Multiplications, 4) Divisions, 5) Powers, or 6) the taking of Roots to this Collection.

EXAMPLE ALGEBRAIC EXPRESSIONS (32)

 

POLYNOMIALS  Write General Polynomial Form on the BOARD

a_nxⁿ + … + a_o where n is a non-negative integer and each coefficient is a REAL NUMBER.  If a_n¹0 then the polynomial is said to be of DEGREE n.

TERMS ¾ each combination of a coefficient and a x to a power is called a TERM of the polynomial.  If the coefficient is 0 then it is usually not written.  The Coefficient of the highest power in the polynomial is called the LEADING coefficient.

 

EQUALITY ¾ two polynomials are EQUAL if they have the same TERMS – i.e. the same coefficients on the same powers – if all the coefficients are zero the polynomial is called the ZERO POLYNOMIAL.  The degree of the Zero Polynomial is said to be UNDEFINED.  The reason for that is that a CONSTANT term – a non-zero real number standing alone is terms a CONSTANT Polynomial.

 

DO SOME EXAMPLES.  See page 34 for 1) NON-POLYNOMIALS

then do examples of

2) ADDING and SUBTRACTING Polynomials – this is done by GATHERING COMMON TERMS (i.e. terms with the same POWER) and then adding or subtracting the coefficients.

3) MULTIPLYING BINOMIALS (34)

4) MULTIPLYING POLYNOMIALS (35)

 

5) DIVIDING A POLYNOMIAL BY A MONOMIAL (36)

 

COMMON PRODUCT FORMULAS  (36)

EXAMPLES: Using Product Formulas

FACTORING – if a polynomial results from a product of other polynomials, then recovering the parts of the product (the FACTORS) is called Factoring. 

Factoring is the process of expressing a sum of terms as a PRODUCT.

Usually we are interested in NONTRIVIAL factors – whole numbers or lesser degree polynomials.

IRREDUCIBILITY – a Polynomial with coefficients in some set S is PRIME or IRREDUCIBLE over S if it cannot be written as a Product of two polynomials of positive degree with coefficients in S.   A polynomial can be Irreducible over one set, say the rational number and not irreducible over a different set, ex. the reals. x²-2 is irreducible over the Rationals and reducible over the Reals.

FACTORING FORMULAS – (38) see table on page.

EXAMPLES – Difference of 2 Squares, Sum and Difference of two Cubes

 

TRIAL AND ERROR (40)  -- (ax+b)(cx+d)=acx²+(bc+ad)x +bd.

-- figuring them out – do some examples

 

FACTORING POLYNOMIALS

See examples Example 9 (41)

Example 10 (42)

 

If time permits do some example problems from the end of the chapter.