Difference Between Positive and Negative Integers

Zero IS an integer, but it is neither postive or negative. It is sometimes called "non-negative," but it is not positive. Positive integers are also called "natural" or "counting" numbers. You use them to count: 0,1,2,3,4,5... Negative integers are the "opposite" of counting numbers. The opposite of 1 is minus(negative) 1. The opposite of 2 is minus 2. And so on. These negative numbers are steps backwards on a number line.
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An integer is any whole number that doesn't contain anything beyond (to the right of) the decimal point. Positive and Negative are adjectives that describe the integer. Positive means greater than or equal to 0 (eg. 0,1,2,3,4,5...) and negative means less than or equal to 0 (eg. 0,-1,-2,-3,-4,-5...). Negative integers are preceded with a "-" sign except for 0 because -0=0.

The set of positive integers and the set of negative integers when used by themselves act the same way under addition (they are just spelt a little differently)
3+3=6
-3-3=(-3)+(-3)=(-6)
They act differently under multiplication though.
4*5=20
(-4)*(-5)=20 (not -20)
but (-4)*5=4*(-5)=-20


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Integers


The integers are a special set of numbers often called whole numbers.
Whole numbers, however, consist of the counting numbers 1,2,3,4,5,6,7,8,9,10,11,..
along with zero. The integers is a larger set because it includes the negatives of
the counting numbers-1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, ...
Integers form an algebraic group under addition and are closed under multiplication.
The integers are the natural numbers (also called counting numbers) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ... and their negatives -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, ... and 0.
Zero is the additive identity and for any integer x its additive inverse is -x (i.e. x - x = x + (-x) = (-x) + x = 0).
Also -(-x)=x