We've seen that natural numbers are closed under addition. Determine the closure of natural numbers under the other three operations, and give examples to support your answers.

The natural numbers are the counting numbers.

The set of natural numbers consists of 1, 2, 3, 4, 5, ...

Zero is not part of the natural numbers.

You need to know what it means for natural numbers to be closed under addition. That means that if you add any two natural numbers, you get a natural number. Natural numbers are closed under addition since the sum of any two natural numbers is a natural number. You cannot find any pair of natural numbers whose sum is not a natural number. 5 + 6 = 11; 10 + 12 = 22; 40 + 50 = 90. Al the numbers above are natural numbers.

Now think of multiplication. If you multiply any two natural numbers, will the answer always be a natural number? The answer is yes. That means that natural numbers are closed under multiplication. For example, 4 * 2 = 8; 10 * 20 = 200; 15 * 3 = 45. All numbers above are natural numbers.

What about subtraction? If you subtract a natural number from another natural number, will the difference always be a natural number?

Look at 8 - 18 = - 10. The difference, - 10, is not a natural number, so the natural numbers are not closed under subtraction.

Finally, think of division. If you divide a natural number by another natural number, the quotient may or may not be a natural number.

For example, 24/6 = 4. Here the quotient is a natural number,

but 4/24 = 1/4. 1/4 is not a natural number,

so with division, you may not get a natural number, so the natural numbers are not closed under division.