A journey that crosses zero

Start at −3 and apply +5. Then try returning 4 units. Where does each route end?

Em telas pequenas, deslize o gráfico na horizontal para ler os detalhes.

Explore the controls and explain what changed in your own words.

In the study of naturals, we use numbers to count, including zero. Now let’s expand this set with negative numbers and know the integers. Naturals are represented as:

$$\mathbb{N} = \\{0,1,2,3,4,5,\dots\\}$$

Quantity and natural numbers

To talk about integers, we need to talk about negative numbers. When we talk about natural numbers, quantity can be represented by these numbers:

  • 2 apples
  • 5 children
  • 3 cars
  • Among others
2 apples 5 children 3 cars

This idea has been intuitive since before Christ, that is, we have always had the notion of quantity.

Negative Numbers

In everyday life, there is also a need for a representation of negative numbers. Temperature is an example of a negative number.

For example, in Japan, Russia, among other countries, in winter the temperature can be negative.

Saporo Japan Winter in Japan
Moscow river Winter in Russia

Definition of the integer

As an introduction to integers, we need to have an idea about opposites.

Heaven and Hell Light and Dark

These are examples of opposites, but for qualities (qualitative). To talk about opposites, but with quantities (quantitative), it is necessary to talk about whole numbers.

Firstly,

  • given a natural number $n,$
  • where this number belongs $(\in)$
  • to the set of natural numbers $(\mathbb{N})$.
$$n\in \mathbb{N} = \\{0,1,2,3,4,5,\dots\\}$$

We define negative numbers as:

  • $-n = n_\star$
  • the opposite in the natural number $n$
  • with $n$ non-zero $(n\neq 0)$

In this case, zero will function as the center of mass, or the point of balance between positive and negative numbers.

In this way, we obtain

$$n + n_\star = 0 = n_\star + n$$

for every natural number $n\in \mathbb{N}.$

Therefore, when we join $(\cup)$ the positive and negative numbers, we obtain a set:

$$\\{\dots, -3,-2, -1\\} \cup \\{0\\}\cup \mathbb{N}$$
Negative, zero and positive integers.
Negatives, zero and positives form ℤ.
Ampliar figura para ler os detalhes
Negative, zero and positive integers.

Definition: The set of integers is given by:

$$\mathbb{Z} = \\{\dots, -3,-2, -1, 0, 1, 2, 3, \dots\\}$$

Therefore, the representation of natural numbers and integers is given by:

Set of naturals with zero contained in the integers.
We use ℕ with zero: every natural is also an integer.
Ampliar figura para ler os detalhes
Set of naturals with zero contained in the integers.

This means that the set of natural numbers is contained in the set of integers.