The Mysterious World of Number Identity… What’s In A Number? The Mysterious World of Number Identity…

Categories of Numbers in the REAL Number System Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers

Natural Numbers Are the counting numbers, or all positive numbers, that mean we exclude negative numbers and zero {1, 2, 3, 4, 5, 6, 7, 8, …}

Whole Numbers All of the counting numbers and zero. {0, 1, 2, 3, 4, 5, 6, 7, …}

Integers Are all of the natural numbers, their opposites (negative numbers) and zero. {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Rational Numbers Numbers that can be expressed as a fraction (a/b). This set includes the integers, terminating decimals, and repeating decimals. Some examples: 2 = 2/1 3 ¼ = 13/4 -0.25 = -25/100 1/3 = 0.33333333333333333333333

Irrational Numbers Numbers that CANNOT be expressed as a fraction of integers. In decimal form, they are the numbers that go on forever without a repeating pattern. Some examples: √2 = 1.4142… π = 3.1415… 45.9492…

Venn Diagram of REAL Number System Rational Numbers Integers Whole Natural Irrational Numbers

Tree Diagram of Real Number System 0.5, 1.4, 0.256 0.3, 0.45 0.6392518… Π, √2 1/5, 4/11, 12/3 -5, -80 1, 2, 3…

Classify each number as natural, whole, integer, rational, or irrational. Write as many as apply. 7.4569594… -5 ¾ -79 3 √16

The Real Number System zero The natural numbers are the counting numbers, without _______. Whole numbers include the natural numbers and ____________. Integers include all whole numbers and their ________________. Rational numbers are real numbers that can be written as a _____________ where a and b are integers and b ≠ 0. Any rational number can be represented as a terminating or a repeating ___________. Irrational numbers are any real numbers that are not _________. zero opposites   decimal rational

Set A collection of objects. Set Notation { } Natural numbers Counting numbers {1,2,3, …} Whole Numbers Natural numbers and 0. {0,1,2,3, …} Integers Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …} Vocabulary Rational Number A real number that can be expressed as a ratio of integers (fraction) Irrational Number Any real number that is not rational. Real Numbers All numbers associated with the number line.

Translating English to Maths sum of two numbers difference between two numbers The product of two numbers the quotient of two numbers is a + b a - b ab b a =

B I M D A S ORDER OF OPERATIONS ultiplication rackets ubtraction When there is more than one symbol of operation in an expression, it is agreed to complete the operations in a certain order. A mnemonic to help you remember this order is below. Complete multiplication and division from left to right Apply Indices Complete addition and subtraction from left to right Do any simplifying possible inside of brackets starting with innermost brackets and working out rackets ultiplication ndices ivision ddition ubtraction B I M D A S

BIMDAS BIMDAS BIMDAS BIMDAS BIMDAS complete addition and subtraction, left to right complete multiplication and division, left to right indices – apply the indice now brackets – combine these first BIMDAS BIMDAS BIMDAS BIMDAS BIMDAS

Properities of real numbers:

COMMUTATIVE PROPERTY The operations of both addition and multiplication are commutative, that not applied for division or subtraction. When adding, you can “commute” or trade the terms places When multiplying, you can “commute” or trade the factors places

ASSOCIATIVE PROPERTY The operations of both addition and multiplication are associative When adding, you can “associate” and add any terms first and then add the other term. When multiplying, you can “associate” and multiply any factors first and then multiply the other factor.

DISTRIBUTIVE PROPERTY The operation of multiplication distributes over addition The distributive property also holds for a factor that is multiplied on the left.

CAUTION: Remember that the value for a and/or b could also be positive or negative. Rules of Signs A positive times a negative is A negative times a positive is NEGATIVE NEGATIVE The negative of a negative POSITIVE A negative divided by a positive is NEGATIVE A positive divided by a negative or A negative divided by a negative is POSITIVE

You are familiar with the following type of numerical expressions: 12 + 6 3 (12) 6 (3 + 2) 15 - 4 (6)

What is a variable? In the expression 12 + B, the letter “B” is a variable. A variable is a letter or symbol that represents an unknown value.

Algebraic Expressions When variables are used with other numbers, parentheses, or operations, they create an algebraic expression. a + 2 (a) (b) 3m + 6n - 6

What are coefficients? A coefficient is the number multiplied by the variable in an algebraic expression. Algebraic Expression Coefficient 6m + 5 6 8r + 7m + 4 8, 7 14b - 8 14

What is a term? A term is the name given to a number, a variable, or a number and a variable combined by multiplication or division. Algebraic Expressions Terms a + 2 a, 2 3m + 6n - 6 3m, 6n, - 6

What are constants? A constant is a number that cannot change its value. In the expression: 5x + 7y - 2 the constant is - 2.

Figure it out! Identify the terms, coefficients, and constants. 1. 12a - 6b + 4 2. 4x - 2y 3. c - 32 4. 3x + 2

Writing Algebraic Expressions You can translate word phrases into variable expressions. Examples: Three more than a number = x + 3 The quotient of a number and 8 = y/8 Six times a number = 6 x n or 6n 15 less than a number = z – 15 The quotient of 30 and a number plus 10 = 30/x + 10.

The Laws of Exponents

Exponents exponent Power base 53 means 3 factors of 5 or 5 x 5 x 5

#1: Exponential form: The exponent of a power indicates The Laws of Exponents: #1: Exponential form: The exponent of a power indicates how many times the base multiplies itself. n factors of x

#2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS! So, I get it! When you multiply Powers, you add the exponents!

#3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS! So, I get it! When you divide Powers, you subtract the exponents!

Try these:

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#4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents! So, when I take a Power to a power, I multiply the exponents

#5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent. So, when I take a Power of a Product, I apply the exponent to all factors of the product.

#6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent. So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.

Try these:

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#7: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent. So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent. Ha Ha! If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!

#8: Zero Law of Exponents: Any base powered by zero exponent equals one. So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.

Try these:

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Recognize Monomials