Operations on Integers
- When two positive integers are added, we get a positive integer; 9 + 2 = 11.
- When two negative integers are added, we get a negative integer; -2 + (-3) = -5.
- When a positive and a negative integer are added, the sing of the sum is always the sing of the bigger number of the two, without considering their sign 55 + -25 = 30 and -55 + 25 = -30.
- The additive inverse of any integer a is -a , and the additive inverse of (-a ) is a . Integer (-12) = Additive Inverse (12).
- Subtraction is the opposite of addition and therefore, we add the additive inverse of the integer that is being subtracted, to the other integer.Hence, 23 - 43 =23 + Additive Inverse of 43 + 23 + (-43) = -20.
- The product of a positive and a negative integer is a negative integer.
- If the number of negative integers in a product is even, then the product is a positive integer, and if the number of negative integers in a
- product is odd, then the product is a negative integer.
- Division is the inverse operation of multiplication.
- The division of a negative integer by a positive integer results in a negative integer.
- The division of a positive integer by a negative integer results in a negative integer.
- The division of a negative integer by a negative integer results in a positive integer.
- For any integer p, p multiplied with zero is equal to zero multiplied with p, which is equal to zero.
- For any integer p, p divided by zero is not defined, and zero divided by p is equal to zero, where p is not equal to zero.
Properties of Integers
- Integers are closed under addition and subtraction. That is, a + b and a - b are again integers, where a and b are any integers.
- Addition is commutative for integers, i.e., a + b = b + a, for all integers a and b .
- Addition is associative for integers, i.e., (a + b) + c = a + (b + c), for all integers a, b and c.
- Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a, for every integer a .
- Integers are closed under multiplication. That is, a × b is an integer, for any two integers a and b .
- Multiplication is commutative for integers. That is, a × b = b × a for any integers a and b .
- The integer 1 is the identity under multiplication, i.e., 1 × a = a × 1 = a, for any integer a .
- Multiplication is associative for inters, i.e.,(a × b) × c = a × (b × c), for any three integers a, b and c.
- Distributive property of multiplication over addition: For any integers a, b, c we have a × (b + c) = a × b + a × c.
- Distributive property of multiplication integers a, b, c we have a × (b - c) = a × b - a × c.
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