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Questions and Answers
An integer is a whole number, either ______, negative, or zero, without a fractional part.
positive
The ______ property states that the order of integers does not change the result of their addition or multiplication.
commutative
The ______ property states that the order in which integers are added or multiplied does not change the result.
associative
The multiplication of an integer by the sum of two integers is equal to the sum of the multiplication of the integer by each of the two ______.
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______ integers are integers that are greater than zero.
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The sum of two integers is always a ______.
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The result of dividing one integer by another is not always a ______.
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Integers are used in real-world applications such as counting and ______ quantities.
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Study Notes
Definition
- An integer is a whole number, either positive, negative, or zero, without a fractional part.
- It is a number that can be expressed without a decimal or fractional component.
Properties
- Commutative Property: The order of integers does not change the result of their addition or multiplication. For example, 2 + 3 = 3 + 2 and 2 × 3 = 3 × 2.
- Associative Property: The order in which integers are added or multiplied does not change the result. For example, (2 + 3) + 4 = 2 + (3 + 4) and (2 × 3) × 4 = 2 × (3 × 4).
- Distributive Property: The multiplication of an integer by the sum of two integers is equal to the sum of the multiplication of the integer by each of the two integers. For example, 2 × (3 + 4) = 2 × 3 + 2 × 4.
Types of Integers
- Positive Integers: Integers that are greater than zero. For example, 1, 2, 3, ...
- Negative Integers: Integers that are less than zero. For example, -1, -2, -3, ...
- Non-Positive Integers: Integers that are less than or equal to zero. For example, 0, -1, -2, ...
- Non-Negative Integers: Integers that are greater than or equal to zero. For example, 0, 1, 2, ...
Operations on Integers
- Addition: The sum of two integers is always an integer. For example, 2 + 3 = 5.
- Subtraction: The difference of two integers is always an integer. For example, 5 - 3 = 2.
- Multiplication: The product of two integers is always an integer. For example, 2 × 3 = 6.
- Division: The result of dividing one integer by another is not always an integer. For example, 5 ÷ 2 = 2.5 (not an integer).
Real-World Applications
- Counting and measuring quantities
- Representing temperatures above or below zero
- Denoting positive or negative changes in quantities
- Solving problems involving profit and loss, or gain and debt
Definition of Integers
- An integer is a whole number, either positive, negative, or zero, without a fractional part.
- It can be expressed without a decimal or fractional component.
Properties of Integers
- The commutative property of integers states that the order of integers does not change the result of their addition or multiplication.
- The associative property of integers states that the order in which integers are added or multiplied does not change the result.
- The distributive property of integers states that the multiplication of an integer by the sum of two integers is equal to the sum of the multiplication of the integer by each of the two integers.
Types of Integers
- Positive integers are greater than zero, e.g., 1, 2, 3, ...
- Negative integers are less than zero, e.g., -1, -2, -3, ...
- Non-positive integers are less than or equal to zero, e.g., 0, -1, -2, ...
- Non-negative integers are greater than or equal to zero, e.g., 0, 1, 2, ...
Operations on Integers
- The sum of two integers is always an integer, e.g., 2 + 3 = 5.
- The difference of two integers is always an integer, e.g., 5 - 3 = 2.
- The product of two integers is always an integer, e.g., 2 × 3 = 6.
- The result of dividing one integer by another is not always an integer, e.g., 5 ÷ 2 = 2.5 (not an integer).
Real-World Applications of Integers
- Counting and measuring quantities
- Representing temperatures above or below zero
- Denoting positive or negative changes in quantities
- Solving problems involving profit and loss, or gain and debt
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Description
Learn about the definition and properties of integers, including commutative and associative properties. Understand the basics of integers and their behavior in mathematical operations.
