Real Numbers and Properties

Chapter 1: Real Numbers

• Classifies different types of real numbers
• Commutative and associative properties get used
• Simplify expressions utilize the distributive property
• Recognize and apply identity and inverse properties regarding addition and multiplication
• Applies all properties of zero
• Simplify expressions, uses, and applications of identities, inverses, and zero
• How to add and subtract fractions
• Converts percents to fractions and decimals
• Converts decimals and fractions to percents

Classification of Real Numbers

• Numbers and symbols represent words and ideas in algebra
• Real numbers, subsets, and the relationships between them are described

Identify Counting Numbers and Whole Numbers

• 1, 2, 3, 4, 5 are the most basic numbers for counting objects in algebra
• Counting numbers are natural numbers
• N = {1, 2, 3, 4, 5, …}
• Mathematics saw a major advancement when zero was discovered
• Adding zero defines a new number set: whole numbers
• W = {0, 1, 2, 3, 4, 5, …} is whole numbers

Introduction to Integers

• It is less than 0 for a number to be negative
• A number line can represent both positive and negative numbers
• A plus sign can be written before a positive number - +2 or +3
• It is standard practice not to include the plus sign
• If there is no sign, the number is positive
• The number line extends infinitely in both directions, indicated by arrows on each end
• There is no greatest positive or smallest negative number

Integers

• Is defined by counting numbers, their opposites, and zero
• Z = {…,-3,-2,-1,0,1,2,3,…}

Rational and Irrational Numbers

• Examines rational numbers

Rational Numbers

• (Q) is a number that is in p/q form
• p and q are integers
• q ≠ 0
• Q = {p/q: p and q are integers and q ≠ 0}
• An integer can be a rational number if expressed as a ratio
• An integer can be written as a fraction with a denominator of one to decide
  • 3 = 3/1,-8 = -8/1, and 0 = 0/1
• All the whole and counting numbers are also integers, so they are rational
• A decimal can be rational - 7.3 = 73/10
• A ratio of integers or decimal that terminates, [ex: 4.275] or repeats [ex:2.757575…] are rational

Irrational Numbers

• (I) are any decimals that do not stop or repeat because they cannot be a ratio of integers
• π = 3.141592654
• Square roots of numbers that are not perfect squares are decimal representations that are irrational
  • √5 = 2.236067978
• Irrational numbers, I are numbers that cannot be expressed as a ratio of two integers. Decimal forms of these numbers are non-repeating, non-terminating
• For example:
  • 0.583 = 0.583333 is a repeating decimal and, therefore, a rational number
  • 0.475 is a terminating decimal and, therefore, a rational number
  • 3.605551275… contains the ellipsis symbol ( … ) which means that the number does not terminate, or repeat. So, 3. 605551275… is an irrational number
• Square roots of perfect squares are always rational whole numbers
• However, the decimal forms of square roots are non-terminating, or non-repeating

Classify Real Numbers

• Counting numbers, whole numbers, integers, and rational numbers all relate
• Irrational numbers are different
• Combining rational with irrational numbers is how to obtain the set of real numbers

Real Numbers

• Refer to numbers that are either rational or irrational

Properties of Real Numbers

Closure Property of Addition

• If a and b are real numbers, then a + b is also a real number

Closure Property of Multiplication

• If a and b are real numbers, then a⋅b is also a real number

Commutative Property of Addition

• If a and b are real numbers, then a + b = b + a

Commutative Property of Multiplication

• If a and b are real numbers, then a⋅b = b⋅a

Associative Property of Addition

• If a, b, and c are real numbers, then (a + b) + c = a + (b + c)

Associative Property of Multiplication

• If a, b, and care real numbers, then (a⋅b)⋅c = a⋅(b⋅c)

Additive Identity

• Identity property of addition: For any real number a, a + 0=a
• Zero is the additive identity

Multiplicative Identity

• Identity property of multiplication: For any real number a, a⋅1=a
• One is the multiplicative identity

Inverse Property of Addition

• For any real number, a, a + (-a) = 0
• –a is the additive inverse of a

Inverse Property of Multiplication

• For any real number a ≠ 0, a⋅1/a = 1
• 1/a is the multiplicative inverse of a

Distributive Property

• If a, b, and c are real numbers, then a⋅(b + c) = a⋅b + a⋅c

Properties 0f Zero

• For any real number, a, a⋅0 = 0 and 0⋅a=0
• 0/a = 0 for any real number, a ≠ 0
• a/0 is undefined for any real number a

Fractions, Decimals, Ratios and Percent

• There are types of numbers describing parts
• Fractions serve a huge role in everyday life and algebra

Fractions

• Shows how many portions of the whole are present
• The denominator, b, represents equal portioned parts of the whole
• The numerator, a, represents included parts
• Division results in the denominator - b ≠ 0

Fractional Form

• Has a as numerator and b as denominator
• Expresses the relationship when a is divided by b

Proper Fractions and Mixed Numbers

• Fractions can be proper or improper

Proper Fraction

• Exists when numerator is less than denominator - a < b

Improper Fraction

• Exists when numerator is greater than or equal to denominator - a ≥ b
• An improper fraction can be translated into a mixed number

Mixed Number

• Whole number followed by a fraction

Multiples and Factors

Multiple

• Is a specific number's product alongside a counting number - 3, 6, 9, and 12 are the multiples of 3

Divisibility

• If m is a multiple of n, m is divisible by n

Factors

• If a ⋅ b = m, a and b are the factors of m, where m is a product of a and b

Steps to Finding a Counting Number's Factors

• Divide the number by every counting number sequentially. When the quotient is lower than the divisor, conclude
• When the divisor & quotient have a counting number, they create a factor pair
• When the quotient doesn't contain a counting number, recognize that the divisor is not a factor
• List every factor pair
• Write each factor out, order smallest to largest

Prime and Composite Numbers

• Numbers have various factors like 72
• Other numbers like 7, only have two factors
• With only two factors, a number is prime, and if there are more than two, the number is composite
• The number 1 is different, and has one factor

Prime Numbers and Composite Numbers

• Number which has a value over 1, and only factors for 1 and/or themselves is a prime number
• Counting, non-prime numbers, are composite numbers
• Examples:
  • Values of 2, 3, 5, 7, 11, 13, 17, and 19 are prime numbers below 20 Values equal to or below 25 for composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, and/or 25

Equivalent Fractions

Equivalent

• Possess equal value

• To work with fractions, understand writing fractions having an equal value

• There are a number of ways to write an equal valued fraction

• A simplified fraction is one that doesn't have common factors, other than one, in the numerator and denominator

Fraction Multiplication

• Exists with a, b, c, and d numbers, in which b ≠ 0 and d ≠ 0
• Then multiply a/b and c/d = ac/bd
• Numerators and denominators, must be multiplied

Reciprocals

• a/b’s reciprocal is b/a, where a ≠ 0, & b ≠ 0
• Multiplying the number and its reciprocal yields 1

Fraction Division

• Occurs with numbers a, b, c, and d, so that b ≠ 0, c ≠ 0, and d ≠ 0 Then a/b ÷ c/d = a/b ⋅ d/c = ad/bc

Add and Subtract Fractions with Common Denominators

• Apply numbers a, b, and c, where c ≠ 0
  • a/c + b/c = a+b/c, and a/c - b/c = a-b/c

Add and Subtract Fractions with Different Denominators

• Equivalent fractions have common denominators for adding fractions
• Finding the lowest denominator for two fractions is the least common range, i.e. the LCM, or lowest common multiple, of the denominators

Steps to obtaining the lowest common denominator (LCD) of two Fractions:

• Decompose both denominators to primes
• List each prime, lining up identical primes to form columnar relations
• Copy each column downwards
• Multiply factors together to yield the LCM to act as the denominators' LCD

Decimals

• Steps related to decimal additions are:
  • Align decimals along a line as part of writing each number downwards
  • Append placeholder zeroes if needed
  • Perform subtraction or additional in practice as though there were whole numbers
• Assign the decimal to its result

Ratios

• Is a unit of comparison between two numbers or quantities, and can be measured with matching units
• The ratio between a & b numbers is written a to b, a/b, or a: b

Percent

• A certain relationship that is expressed with any figure in coordination to its value out of 100
• The ‘%’ represents "out of 100."

Chapter 2: Exponents and Radicals

• Covers exponent laws and ways to simplify expressions and how to manipulate ratios, fractions, percentages and decimals

Integer Exponents Definition

• Square it, or elevate to a strength of 2 when multiplying a number by itself a² = a a = a used as factor n times
• a" read as the nth power of an a, with a as base and n as exponent in this notation

Integer Exponents

Product Rule of Exponents

• This means that when multiplying exponential expressions are multiplied with the same base, the base is written with the exponents added $$a^m * a^n = a^{m+n}$$

Quotient Rule of Exponents

• Allows simplification of numbers that are dividable with the same base and other exponents, where the base is not zero, is the positive m and n integers and power rule all state that $${a^m \over a^n}= a^{m-n}$$

Types of Exponents Rules

Positive

• You add exponents

Quotient

• You subtract exponents

Power Rule of Exponents

• Care is needed distinguishing power rule vs. product, and when using power rule, the exponential notion is raised to a power, and exponents need to be multiplied $$a^m$$" = "a^{mn}$$

Zero Exponents

• Use the zero exponent rule to simplify with an expression of 1: t° = 1, with the caveat that a non-zero, real number expression of 0° is not permissible

Rule of Negative Exponents

• The reciprocal becomes clear from here in simplifying the expression to its negative exponent, where if m < n, then the negative rule of the expression simplifies to its reciprocal to then divide the expression for its reciprocal value- For non-zero, real value of a, and natural form n, then the negative exponents rule states: $$a^{-n} = {1 \over a^n}$$

Rule of a Product

• The use of a product rule is applied then and states that the power needs factoring into its product
• the form on the product powers that is also the product that are then factored into their particular factors $$(ab)^n = a^n b^n$$

Rule of a Quotient

• The powers that factors must follow as the powers share quotient's ()n = /

• For any real non-zero figures a and b along with non-zero n’s quotient power rule, the following statements stay compliant ()n= /

• Reminder to simply expressions, it means expression rewrite combining their expressions or exponents

• The exponents' laws must be followed to simplify

• To simply expression, that means rewriting combining their terms or exponents

• The rules of exponents needs to be followed for simplification

Rational Exponents

• Indicates that are numerators, denominators as square roots - for example of writing square root over 16, it can also be 16½
• The square root form can be a way to describe one writing of 8 or using the exponential of 81/3.
• The skills of working with rational exponents is important

Definition

• Rational Exponents indicate root on denominator and square function in numerator & forms for expression writing whether its variable symbol etc., can hold square notation $$ a^{m \over n}=( \sqrt[n]{a})^m = \sqrt[n]{a^m} $$

Simplify a

3

• Write 1253 = (53)2/3
• Distribute to get 52 = 25

Square Root Functions

• The following list contains Properties of the square root
  • Expression list If an and m are rational number

Product

Multiplying with common notation an + an = Expression, if its the value
Is divided then

Zero Exponential Value

a° to be 1 ( a cannot register as zero) Negative exponents can inverse fractions to show division, the rules must account for rational number cases

Adding Expression types using Rational Expression

• The numerator is identified, first for adding any rational expression types .

Multiplication types using

• Example with solving to factor for what square to

Formula Application with Rational type

• If a side equals 4 and its for square then . An = ( An expression type.

Quadratic Form

• ax² + bx+ c = 0 where a can 2/3 etc.

Simplification

• Use the product law to expand term from product powers for simplification & follow new defined equation notations

Real & Product Rules

• The solution that can solve in product/square forms

Multiplication/Quotient Types

• Use Zero properties where a^0. And inverse exponents as well and square, do not always perform the operations

Radicals

Integer Roots

• n indicates root for a number = ( nth root form a

To solve root equation with exponents, follow this form ( nth power ) to be nth power√a .

Section 2.3: Radical Conversion For Expressions

• (An) power value shows exponential form.

Expression Equation

Square value a for to √(sqrt) An = if √(sqrt) a >=two , here for an equal number square value.

Integer Root- Finding Formula (nth)

• The power is, n, ( where all (√a) power value , ( a ≥ 2), then a to power n , a square symbols for number cases

Power/Rational Exponent

• All value type of roots from an power notation

Definition Of Va

• Value is an odd sum and value or an even number
• Follow real square process power notation’s

Power Exponents

A number > 0 for a then value symbol square types’ notation’s, and a-number <0 , then’s the square root not an real.

Zero Notation

• Where you then be in need, for a to need

For Numbers With Negative

• Remember for symbol- square, what to do, it,

Number Set Value

• Always has variable sign.

Simplified Square Notation

• An notation, √a is named simplified assuming its number square power for m notation.

Simplified Exponents

• Real type numbers has a square as m > 2 . a, is taken, value can’t have root for m notation For a √8 example = 2 root notation for notation, (8), root notations is’t perfect squares factors with an integer, real notation

• There are few properties that follow simplification

Adding Product

• The simplified expression that contains product radicals for simplification to rewrite the real number notation, (b) to has symbol, = with An B notation with + notation - a can't equal > 2

Properties - Roots

• A √a = real, square notation numbers.

Chapter 3: Polynomials

• Introduction to how you manipulate, express numerical’s and other polynomials

Term

• Constant & multiple variables

Coefficient

• Term Multiple values to has variables notation as terms

Likenesses

• If expressions can simplification

Polynomial equation

• Exponent determines expression power

To Determine Notation Degrees

The square of a power is then terms notation symbol value for all types of notation for a one or to, then variables terms

For A monomial

• Can the value to has multi variable ax^Mm which a equals notation for a to number range

Notation Square

• Notations with expressions & to multiple power

Rationality Polynomial

• The degree expressions is what decides degrees

Zero

Values that can be in power with constant notation

Type

• Find the variables in form or express powers from polynomial, check variables for monomial, trinomial etc., to test power

For Addition/ Subtraction

The first step is the terms and notation’s, and the expression, follow step to isolate factor , using all to rules for power symbol from expression with to law, and with each step that does so express notation & powers