Questions and Answers
What is the identity element for addition in the set of integers?
Which statement about subtraction of integers is true?
When adding two integers of different signs, what is the correct procedure?
Where are negative integers located on the integer number line?
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What does the closure property of integers state?
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How do you perform the subtraction of two integers using their opposites?
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If integers are represented on a number line, how do you illustrate addition?
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What is the result of adding two integers with the same sign?
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Study Notes
Properties of Integers
- Definition: Integers include positive numbers, negative numbers, and zero.
- Closure Property: The sum or difference of two integers is always an integer.
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Commutative Property:
- Addition: a + b = b + a
- Subtraction is not commutative: a - b ≠ b - a
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Associative Property:
- Addition: (a + b) + c = a + (b + c)
- Subtraction is not associative: (a - b) - c ≠ a - (b - c)
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Identity Element:
- Addition: The identity is 0 (a + 0 = a).
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Inverse Element:
- For every integer a, there exists -a such that a + (-a) = 0.
Addition and Subtraction of Integers
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Addition Rules:
- Same sign: Add the absolute values and keep the common sign (e.g., 3 + 5 = 8, -3 + (-5) = -8).
- Different signs: Subtract the smaller absolute value from the larger one and keep the sign of the integer with the larger absolute value (e.g., 5 + (-3) = 5 - 3 = 2; -5 + 3 = -5 - 3 = -2).
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Subtraction Rules:
- To subtract an integer, add its opposite (e.g., a - b = a + (-b)).
- Example: 4 - 2 = 4 + (-2) = 2; -3 - 2 = -3 + (-2) = -5.
Integer Number Line
- Definition: A visual representation of integers on a straight line.
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Structure:
- Origin: The point at 0.
- Positive integers: Located to the right of 0 (1, 2, 3, ...).
- Negative integers: Located to the left of 0 (-1, -2, -3, ...).
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Properties:
- Each point on the line corresponds to an integer.
- The distance between any two consecutive integers is equal (1 unit).
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Using the Number Line:
- To add: Move to the right for positive integers and to the left for negative integers.
- To subtract: Move left for positive integers and right for negative integers.
Properties of Integers
- Integers comprise positive numbers, negative numbers, and zero.
- Closure Property: The result of adding or subtracting two integers is always an integer.
-
Commutative Property:
- For addition, the order does not change the sum: a + b = b + a.
- Subtraction is not commutative: a - b ≠ b - a.
-
Associative Property:
- Addition groups can be rearranged without affecting the result: (a + b) + c = a + (b + c).
- Subtraction does not follow this rule: (a - b) - c ≠ a - (b - c).
- Identity Element: In addition, 0 is the identity element since adding 0 to any integer a yields a (a + 0 = a).
- Inverse Element: Every integer a has an opposite -a such that their sum is zero (a + (-a) = 0).
Addition and Subtraction of Integers
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Addition Rules:
- For integers with the same sign, add their absolute values and maintain the common sign (e.g., 3 + 5 = 8 and -3 + (-5) = -8).
- For integers with different signs, subtract the smaller absolute value from the larger, retaining the sign of the larger absolute value (e.g., 5 + (-3) = 2 and -5 + 3 = -2).
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Subtraction Rules: To subtract, convert to addition by adding the opposite (e.g., a - b = a + (-b)).
- Example: 4 - 2 becomes 4 + (-2) = 2; -3 - 2 becomes -3 + (-2) = -5.
Integer Number Line
- A number line is a visual representation that displays integers on a straight line.
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Structure:
- 0 serves as the origin on the line.
- Positive integers (1, 2, 3,...) are positioned to the right of 0.
- Negative integers (-1, -2, -3,...) are located to the left of 0.
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Properties:
- Each point on the line represents a unique integer.
- The distance between consecutive integers is uniform, always one unit apart.
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Using the Number Line:
- For addition of positive integers, move right; for negative integers, move left.
- For subtraction, move left when subtracting positive integers and right when subtracting negative integers.
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Description
Test your knowledge on the properties of integers including closure, commutative, associative properties, and operations like addition and subtraction. This quiz covers essential rules and concepts crucial for understanding integers and their behavior in mathematics.
