1.1: Introduction to Addition

• Definition: Addition is a basic arithmetic operation that combines two or more numbers to find their total or sum. It is denoted by the plus symbol (+).
• Example: If we have two numbers, $a$ and $b$, the sum of $a$ and $b$ is $a+b$.
• Properties: Addition follows several properties, including commutative, associative, identity, and inverse properties, which govern how addition behaves when combining numbers.

1.2:Addition:

• Closure under Addition: This means that when you add two integers, the result is always another integer. For example, if you add 5 and -3, you get 2, which is also an integer.

• Commutative Property: This property states that the order in which you add two integers doesn’t matter; the result will be the same. For example, 5 + (-3) is the same as (-3) + 5, both resulting in 2.

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• Associative Property: This property means that the way you group three or more numbers when adding them doesn’t change the result. For example, (2 + 3) + 4 is the same as 2 + (3 + 4), both resulting in 9.

• Additive Identity: This property states that when you add zero to any integer, you get the integer itself. For example, 7 + 0 is equal to 7.

1.3: Properties of Multiplication

Multiplication:

• Closure under Multiplication: This means that when you multiply two integers, the result is always another integer. For example, if you multiply 2 and -3, you get -6, which is also an integer.

• Commutative Property: This property states that the order in which you multiply two integers doesn’t matter; the result will be the same. For example, 2 × (-3) is the same as (-3) × 2, both resulting in -6.

• Associative Property: This property means that the way you group three or more numbers when multiplying them doesn’t change the result. For example, (2 × 3) × 4 is the same as 2 × (3 × 4), both resulting in 24.

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• Multiplicative Identity: This property states that when you multiply any integer by 1, you get the integer itself. For example, 7 × 1 is equal to 7.

1.4: Introduction to Division

• Definition: Division is an arithmetic operation that involves splitting a quantity into equal parts or groups. It is the inverse operation of multiplication.
• Example: Suppose we have $a$ divided by $b$, denoted as 𝑎𝑏ba​, where $a$ is the dividend, and $b$ is the divisor. Division is essentially finding out how many times the divisor can be subtracted from the dividend without resulting in a negative number.
• Properties:
  • Division by Zero: Division by zero is undefined in arithmetic. It violates the fundamental property of division, which states that any number divided by zero is undefined.
  • Division of Fractions: Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction.
  • Division of Integers: Division of integers can result in a quotient that is a fraction or a whole number, depending on whether the division is exact or not.
• Quotient and Remainder: Division can be expressed as the quotient and remainder. The quotient represents the number of times the divisor can be subtracted from the dividend, and the remainder is what is left over after the division process.
• Long Division: Long division is a method used to perform division of large numbers manually. It involves a series of steps to divide the dividend by the divisor and obtain the quotient and remainder.
• Applications: Division is used in various real-life scenarios, such as sharing items equally among a group of people, calculating rates and ratios, and solving problems involving proportions.

1.5: Introduction to Algebra

• Definition: Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. It involves solving equations, manipulating expressions, and analyzing mathematical structures.
• Basic Concepts:
  • Variables: Algebra uses letters or symbols to represent unknown quantities, which are called variables. Commonly used variables include $x$, $y$, and $z$.
  • Constants: Constants are fixed values that do not change. They are represented by specific numbers or symbols.
  • Expressions: An algebraic expression consists of variables, constants, and mathematical operations (such as addition, subtraction, multiplication, and division) combined in various ways. For example, $3x+5$ and $2y-7$ are algebraic expressions.
  • Equations: An equation is a mathematical statement that asserts the equality of two algebraic expressions. Equations typically contain an unknown variable and require solving to find the value of the variable that satisfies the equation.
• Operations:
  • Addition and Subtraction: In algebra, addition and subtraction follow the same rules as in arithmetic. Like terms can be combined, and terms with different variables cannot be combined.
  • Multiplication: Multiplication in algebra involves multiplying terms together. The distributive property allows the multiplication of a term by a sum or difference.
  • Division: Division in algebra is similar to division in arithmetic, but it also includes dividing algebraic expressions and dealing with rational expressions.
• Solving Equations: Solving algebraic equations involves isolating the variable on one side of the equation to determine its value. This often requires applying inverse operations to both sides of the equation.
• Graphing Equations: Algebraic equations can be represented graphically on a coordinate plane. The graph of an equation is the set of points that satisfy the equation and often forms a curve or a line.
• Applications: Algebra is used in various fields such as physics, engineering, economics, and computer science for modeling real-world problems, analyzing data, and making predictions. It provides a powerful tool for problem-solving and understanding mathematical relationships.