Addition of integers is the mathematical operation of combining two or more integers to obtain a single integer result. It is a fundamental concept in the study of integers, which are positive and negative whole numbers, including zero.
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The addition of two integers with the same sign (both positive or both negative) results in a sum with the same sign, but the magnitude (absolute value) is increased.
The addition of two integers with different signs (one positive and one negative) results in a sum with the sign of the integer with the greater absolute value.
The addition of an integer and zero results in the original integer, as zero is the additive identity.
Addition of integers is commutative, meaning the order of the addends does not affect the sum.
Addition of integers is associative, meaning that the grouping of the addends does not affect the sum.
Review Questions
Explain the rules for adding two integers with the same sign.
When adding two integers with the same sign, the sum will have the same sign as the addends, but the magnitude (absolute value) of the sum will be greater than the magnitude of the individual addends. For example, adding two positive integers, such as 5 + 3, results in a positive sum of 8. Similarly, adding two negative integers, such as -7 + -2, results in a negative sum of -9.
Describe the process of adding two integers with different signs.
When adding two integers with different signs, the sum will have the sign of the integer with the greater absolute value. To find the sum, the absolute values of the addends are subtracted, and the result takes the sign of the integer with the greater absolute value. For instance, 5 + -3 = 2, as the absolute value of 5 is greater than the absolute value of -3, and the sum takes the positive sign. Conversely, -8 + 4 = -4, as the absolute value of -8 is greater than the absolute value of 4, and the sum takes the negative sign.
Analyze the relationship between the addition of integers and the properties of commutativity and associativity.
The addition of integers exhibits the properties of commutativity and associativity. Commutativity means that the order of the addends does not affect the sum, so a + b = b + a. Associativity means that the grouping of the addends does not affect the sum, so (a + b) + c = a + (b + c). These properties hold true for the addition of integers, allowing for greater flexibility and simplification when working with integer addition. For example, 3 + 5 = 8 and 5 + 3 = 8, demonstrating commutativity, while (2 + 3) + 4 = 9 and 2 + (3 + 4) = 9, demonstrating associativity.