• Integers are simply a set of numbers that include the positive and negative “counting numbers” and zero.

• Integers are denoted by the symbol $ℤ$ or simply $Z$ when writing (a bold capital $Z$)

• You may write a set of integers as:

• Note that the set of integers does not include fractional component; there is no $0.5$, $3\frac{1}{2}$, $2$ etc.

• $\le 0$ simply means that your integer is equal to zero or smaller than zero.

• To refer to specific sets of numbers within cake in terms of the set of integers we can use the following notations:

  • Positive integers:

    • $ℤ_+ = \{+1, +2, +3, +4, +5...\}$, notice that zero is not in the set because zero is neither positive nor negative. It is a neutral number.

  • Negative integers:

    • $ℤ_- = \{-1, -2, -3, -4, -5...\}$, notice that zero is not in the set because zero is neither positive nor negative. It is a neutral number.

  • Non-negative integers:

    • $ℤ_{\le 0}= \{0, +1, +2, +3, +4, +5...\}$, here we are talking about numbers that are not negative therefore zero is included in the set because zero is neither positive nor negative. It is a neutral number.

    • $\le 0$ simply means that your integer is equal to zero or bigger than zero.

  • Non-positive integers:

    • $ℤ_{\ge 0}= \{0, +1, +2, +3, +4, +5...\}$, here we are talking about numbers that are not positive therefore zero is included in the set because zero is neither positive nor negative. It is a neutral number.

You may be able to notice that $ℕ_0= ℤ_{\le 0}$, that is the set of numbers represented by one of the two definitions of a Natural number is the set presented by the non-negative integers.

This, I think, eliminates the need of talking about natural numbers in the first place - since it seems to have two definitions, and the definition changes from person to person country to country, and so on.

So let’s just talk about integers since we have all the notations required to represent the specific sets of numbers within the set of all real numbers without a fractional component.

That would be:

• $ℤ_+ = \{+1, +2, +3, +4, +5...\}$

• $ℤ_- = \{-1, -2, -3, -4, -5...\}$

• $ℤ_{\le 0}= \{0, +1, +2, +3, +4, +5...\}$

• $ℤ_{\ge 0}= \{0, -1, -2, -3, -4, -5...\}$

There are other things like ‘Whole numbers’ which can represent so many different things according to different people. So it is best to avoid using something like ‘Whole numbers’ and simply use integers.

Sometimes, in everyday languages, ‘whole’ numbers typically mean ‘a full number’. For example, 4 is a whole number because 4 can represent 4 whole cakes. 4.5 cannot be described as a ‘whole number’ since, in terms of cake, it would be 4 full cakes and half of a cake.

However, I think you can simply call 4 an integer. Since integers basically represent ‘whole or full numbers’. So why use confusing terms if you can simply use the more clear-cut ones?