Integers

"Dad, can you buy me those butterfly wings?"
"They are seven dollars. How about you use some of your own money?"
"But I don't have my money with me."
"If I buy the wings, you can owe me the money and give it to me when we get back home."
Shortly afterwards my daughter has the butterfly wings and -$7 in her pocket and is thus introduced to negative numbers, and spending on credit.

The first extension we can make to the natural numbers is to go backwards as well as forwards to get the integers. The natural numbers start and zero and count 1, 2, 3 and so on. The integers allow us to count backwards as well. So -1, -2, -3... are integers, but are not natural numbers.

Now we can subtract any two integers and get another integer. We are not restricted like we were for the natural numbers. We can take a larger number from a smaller one and get an answer: a negative number.

And like with the natural numbers we can add and multiply any integer and get another integer. We are still restricted with division, though. 12 divided by 3 is fine, but 13 divided by 3 is a problem for the integers, just as it was for the natural numbers.

Strangely, exponentiation (raising a number to the power of something), which was OK for all natural numbers, is a problem for the integers. If we raise an integer to the power of a negative number we do not get an integer back. What is 3^-1? or 2^-2? Not an integer, that's for sure. So we can only use zero and positive powers in the land of the integers.

The Integers are given the symbol: ℤ