Natural Numbers

Simply put, all whole numbers other than \(0\) are natural numbers.

Natural numbers are very natural to humans because they come from counting objects like apples or sheeps. Natural numbers can be used to estimate your possessions, how much you have. If you raise sheep, for example, you need to put them out to pasture. When they have come back, how can you confirm whether all of them are in the fold? If we do not have numbers, then you may use pebbles or twigs; move pebbles or twigs when the sheep go out or come in. If pebbles or twigs have moved completely, you can notice that you have all sheeps. Furthermore, people began to write numerals instead of moving pebbles or twigs. We can see many ancient numerals which resemble the shape of pebbles or twigs. Now we have many numerals such as Mayan, Chinese (一, 二, 三, 四, ...), Roman (Ⅰ, Ⅱ, Ⅲ, Ⅳ, ...), and Hindu-Arabic numerals (0, 1, 2, 3, 4, ...). Numerals are commonly used to represent natural numbers. The following figure shows some Mayan numerals.

Mayan Numerals

The set \(\mathbb{N}\) of natural numbers is the set of one, two (one more than one), three (one more than two), .... Some people may include zero in \(\mathbb{N},\) but herein by \(\mathbb{N}\) let's begin at one. The formation of \(\mathbb{N}\) comes from an operation; addition \((+).\) Instead of adding by one, we may add more than that. When A has two apples and B has three apples, A and B have five apples in total; likewise we can represent merge or increment in possession. We did not stop here. We can multiply \((\times)\) natural numbers. When we have four sets of five apples, we can multiply four and five \((4 \times 5)\) instead of summing four fives \((5 + 5 + 5 + 5).\)

When we compare or remove our possessions, we need an operation; subtraction \((-).\) The difference between two natural numbers is obtained from the larger one subtracted by the smaller one, and we can tell how large a gap we have. The division \((\div)\) is used to distribute apples equally to some people. When we have fifteen apples and three people, a person can take \(15 \div 3 = 5 \) apples if we distributed them equally.

\(\mathbb{N}\) has the following properties:

Closure under addition and multiplication: For all \(a, b \in \mathbb{N} ,\) \[a + b \in \mathbb{N} \quad \text{and}\quad a \times b \in \mathbb{N} .\]

Associativity: For all \(a, b, c \in \mathbb{N} ,\) \[a + (b + c) = (a + b) + c\quad \text{and}\quad a \times (b \times c) = (a \times b) \times c .\]

Commutativity: For all \(a, b \in \mathbb{N} ,\) \[a + b = b + a\quad \text{and} a \times b = b \times a.\]

Distributivity of multiplication over addition: For all \(a, b, c \in \mathbb{N} ,\) \[a \times (b + c) = (a \times b) + (a \times c).\]

Which of the following are natural numbers:

\[\frac{3}{5},\ 1,\ -5,\ \pi,\ 8,\ 2+3i,\ -\sqrt{8},\ 0.3\, ?\]

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Natural numbers are numbers that can be obtained by adding ones. Since \(1=1\) and \(8=1+1+1+1+1+1+1+1,\) the answers are 1 and 8. \( _\square \)

What is \(3 + 5?\)

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Instead of pebbles or twigs, we may fold or unfold our fingers to add 3 and 5:

\[\begin{align} 3 + 5 =& (1+1+1) + (1+1+1+1+1) \\ =& 1+1+1+1+1+1+1+1 \\ =& 8. \ _\square \end{align}\]

What is \(4 \times 6?\)

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We can multiply by a sequence of additions:

\[\begin{align} 4 \times 6 =& 6 + 6 + 6 +6 \\ =& 12 + 6 + 6 \\ =& 18 + 6 \\ =& 24 . \ _\square \end{align}\]

What is \((3 \times 3) + (3 \times 1) + (3 \times 7) + (3 \times 3) + (3 \times 6)?\)

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We can distribute multiplications over more than one addition because the addition of two natural numbers is a natural number. When we have two additions, for example, for all \(a, b, c,d \in \mathbb{N} ,\) we have

\[\begin{align} a \times ( b + c + d) =& a \times ((b + c) + d ) \\ =& a \times (e + d) \\ =& (a \times e) + (a \times d) \\ =& a \times (b + c) + (a \times d) \\ =& (a \times b) + (a \times c) + (a \times d), \end{align}\]

where \(e=b+c.\)

Therefore, we have

\[\begin{align} &(3 \times 3) + (3 \times 1) + (3 \times 7) + (3 \times 3) + (3 \times 6) \\ &= 3 \times (3 + 1 + 7 + 3 + 6) \\ &= 3 \times 20 \\ &= 60 . \ _\square \end{align}\]

Is \(\mathbb{N}\) closed under subtraction?

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No, \(\mathbb{N}\) is not closed under subtraction because \(2-3=-1\) is not a natural number. \(_\square\)

Is division associative in \(\mathbb{N}?\)

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No, division is not associative in \(\mathbb{N}\) because

\[(8 \div 4) \div 2 = 2 \div 2 = 1\]

and

\[8 \div (4 \div 2) = 8 \div 2 = 4\]

are different. \(_\square\)