A **non negative integer** is an integer that that is either positive or zero.

It’s the union of the natural numbers and the number zero. Sometimes it is referred to as Z^{*}, and it can be defined as the as the set {0,1,2,3,…,}.

Z, the set of integers, is defined as {…,-3,-2,-1,0,1,2,3,…}.

## Examples of Non Negative Integers

4, 5, 7, and 9 are all examples of non negative integers, as is 0. -4, -7, -8, and 4.58 are not non negative integers.

## Adding and Subtracting Non Negative Integers

The set of non negative integers is closed under addition and multiplication. What this means is that the sum or product of any two non negative integers will also be a non negative integer. For any *a, b* that are non negative integers, a + b = c and a b = d will also be non negative integers.

The set of non negative integers is not closed under subtraction and division; the difference (subtraction) and quotient (division) of two non negative integers may or may not be non negative integers.

## More Additive Properties of Non Negative Integers

Zero, when subtracted from any non negative integer, gives the integer itself. 0 + a = a for any non negative integer a. So zero is the additive identity. There is only one additive identity in Z^{*}.

Every non negative subtracted from itself is zero. a – a = 0 for every a in Z^{*}.

Addition for non negative integers is commutative; a + b = b + a for every a, b in Z^{*}.

Addition for non negative integers is also associative, a + (b + c) = (a + b) + c for every a, b, and c in Z^{*}

## Multiplying the Non Negative Integer

1, when multiplied by any non negative integer, gives the integer itself. For every a in Z^{*}, 1 · a = a. But 1 is the only multiplicative identity in Z^{*}.

Any number a in Z^{*}, when multiplied by 0, is 0. a · 0 = 0 for every a in Z^{*}.

Multiplication in Z^{*} is both commutative and associative. ab = ba and a(bc) = (ab)c for every a, b, and c in Z^{*}

## Sources

Number Systems Chapter 2

Nonnegative Integers

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