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Algebra I – Preliminaries and Basic Operations

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    Study Guide Algebra I – Preliminaries and Basic Opera Ɵ ons 1. Preliminaries Before you begin learning algebra, it’s important to feel comfortable with some basic math ideas. These ideas come from pre - algebra , and they help you understand how algebra works. In this section, you will review: • Different types of numbers • Ways to write multiplication • Common math symbols Understanding these basics will make learning algebra much easier. Categories of Numbers In algebra, we work with several different kinds of numbers. Each category has its own meaning. Let’s look at them one by one. Natural Numbers (Counting Numbers) Natural numbers are the numbers we use when counting objects. Examples: 1, 2, 3, 4, 5, ... These are sometimes called counting numbers because they start from 1 and go upward. Whole Numbers

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    Study Guide Whole numbers include all natural numbers plus zero . Examples: 0, 1, 2, 3, 4, ... So the only difference between whole numbers and natural numbers is that whole numbers include 0 . Integers Integers include: • positive numbers • negative numbers • zero Examples: ..., − 3, − 2, − 1, 0, 1, 2, 3, ... Integers are important in algebra because they allow us to work with both positive and negative values . Negative Integers Negative integers are numbers less than zero . Examples: − 3, − 2, − 1 These numbers are often used to represent things like loss, decrease, or movement below zero . Positive Integers Positive integers are simply the natural numbers .

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    Study Guide Examples: 1, 2, 3, 4, ... They represent quantities greater than zero . Rational Numbers Rational numbers are numbers that can be written as a fraction . A fraction has the form: a / b where: • a is an integer • b is a natural number (and not zero) Examples: 3/2 7/8 Even numbers like 5 are rational because they can be written as: 5 = 5/1 Rational numbers also include: • terminating decimals (like 0.5 or 0.25) • repeating decimals (like 0.333...) This is because they can all be written as fractions. Irrational Numbers Irrational numbers are numbers that cannot be written as fractions .

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    Study Guide Their decimal form: • never ends • never repeats Examples include: √ 3 π These numbers continue forever without forming a repeating pattern. Even Numbers Even numbers are integers that are divisible by 2 . Examples: ..., − 6, − 4, − 2, 0, 2, 4, 6, ... If a number can be divided by 2 without leaving a remainder, it is even. Prime Numbers A prime number is a natural number that has exactly two factors : 1 and itself. Example: 19 is a prime number because it can only be divided by: 1 and 19 But 21 is not prime because it can be divided by: 1, 3, 7, and 21 Important facts:

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    Study Guide • 2 is the only even prime number • All other even numbers are divisible by 2 and therefore not prime The first 10 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Odd Numbers Odd numbers are integers that cannot be divided by 2 evenly . Examples: ..., − 5, − 3, − 1, 1, 3, 5, ... When divided by 2, they always leave a remainder of 1 . Composite Numbers A composite number is a natural number that has more than two factors . Examples: 4, 6, 8, 9, ... For example: 8 can be divided by: 1, 2, 4, and 8 Since it has more than two factors, it is composite. Square Numbers A square number is the result of multiplying a number by itself . This is also called raising a number to the second power .

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    Study Guide Examples: 2 × 2 = 4 3 × 3 = 9 The first six square numbers are: 1, 4, 9, 16, 25, 36 Cube Numbers A cube number is the result of multiplying a number by itself twice . This means raising a number to the third power . Examples: 2 × 2 × 2 = 8 3 × 3 × 3 = 27 The first six cube numbers are: 1, 8, 27, 64, 125, 216 Ways to Show Multiplication In mathematics, multiplication can be written in several different ways. For example, multiplying 4 and 3 can be written as:

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    Study Guide All of these mean the same thing . Multiplication with Numbers and Variables When one value is a number and the other is a variable, multiplication can be written as: In algebra, we usually write this simply as: 4a This means 4 multiplied by a . Multiplication with Variables When multiplying two variables, the multiplication can be written as: But in algebra, we usually write it simply as: This means a multiplied by b . Common Math Symbols In algebra, you will often see certain symbols. Each one has a specific meaning.

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    Study Guide Here are some of the most common ones. Symbol Meaning = is equal to ≠ is not equal to > is greater than < is less than ≥ is greater than or equal to ≤ is less than or equal to ≯ is not greater than ≮ is not less than ≱ is not greater than or equal to ≰ is not less than or equal to ≈ is approximately equal to These symbols help us compare numbers and expressions in algebra. 2. Quiz: Preliminaries 1. Question An integer is a whole number. Answer Choices • always true • sometimes true • never true Correct Answer : always true Why This Is Correct An integer is defined as a whole number that can be positive, negative, or zero.

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    Study Guide Examples of integers include − 3, − 1, 0, 2, and 7. Because integers do not include fractions or decimals, the statement that an integer is a whole number is always true. 2. Question The decimal form of a rational number terminates. Answer Choices • always true • sometimes true • never true Correct Answer : sometimes true Why This Is Correct A rational number can be written as a fraction (x / y) where x and y are integers and (y ≠ 0). Some rational numbers have terminating decimals, while others have repeating decimals. Examples: • ( ½ = 0.5) (terminating) • ( ⅓ = 0.333…) (repeating) Because rational numbers can either terminate or repeat, the statement is sometimes true. 3. Question If the decimal name of a number never ends but has a repeating pattern, then it is a rational number.

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    Study Guide Answer Choices • always true • sometimes true • never true Correct Answer : always true Why This Is Correct A repeating decimal means the digits continue forever but follow a repeating pattern. All repeating decimals can be written as fractions, which makes them rational numbers. Example: (0.333… = ⅓ ) Since every repeating decimal can be expressed as a fraction, the statement is always true. 4. Question Irrational numbers can be expressed in the form ( x/y ) where x is an integer and y is a natural number. Answer Choices • always true • sometimes true • never true Correct Answer : never true Why This Is Correct

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