Explaining Number Systems to Students: Ideas & Solutions

[roni1]

I need to explain to a student the meaning of number systems.
I try to explain him that it is like a family of numbers that have some proprity.
If one have an idea to expalin him the term, it will be helpful.
I remind note about the term:
A number system can be complex system number, rational system number, irrational system number ...
And more question I get from him is:
How to calculate the execrise:
"(3, 2) * (3, 2) = ?".
How I need to explain it him.
Ideas will be blessed.

 

[I like Serena]

I need to explain to a student the meaning of number systems.
I try to explain him that it is like a family of numbers that have some proprity.
If one have an idea to expalin him the term, it will be helpful.
I remind note about the term:
A number system can be complex system number, rational system number, irrational system number ...
And more question I get from him is:
How to calculate the execrise:
"(3, 2) * (3, 2) = ?".
How I need to explain it him.
Ideas will be blessed.

Hi roni, welcome to MHB!

I suggest to look at e.g. wikipedia for inspiration, which is where such things are explained better than I can.
It explains for instance the term numeral system, which is a writing system to denote a set of numbers.
The set of numbers can be the set of natural numbers, the set of rational numbers, and so on, which are apparently also called number systems.
The system uses specific symbols for writing, such as I, II, III, or 1, 2, 3, with lots of choices although the Arabic numerals 1, 2, 3 have been accepted internationally as the standard.
And the system will use a certain base, such as binary, decimal, or hexadecimal.
Additional characteristics are that each number has a unique representation, and that the notation reflects the algebraic and arithmetic structure of the numbers.

As for calculating "(3, 2) * (3, 2)", that is not uniquely defined.
Can you provide a context?
The most common definitions are:
1. The so called dot product, meaning the result is $3\times 3 + 2\times 2=13$.
2. The component wise product, meaning the result is $(3\times 3, 2\times 2)=(9,4)$.
Without context I would assume that the dot product is intended.

 

[topsquark]

In the process of developing certain number systems I have seen many versions of the notation (a, b).

For example:
1) Given any two integers a, b (b not equal to zero) we can define (a, b) to be the rational number a/b. Then we have \(\displaystyle (a, b) \cdot (c, d) = (ad + bc, bd)\) for addition of rationals and \(\displaystyle (a, b) \cdot (c, d) = (ac, bd)\) for multiplication.

2) Given any two real numbers (a, b) we can define (a, b) to be the complex number a + ib. Then \(\displaystyle (a, b) \cdot (c, d) = (a + c, b + d)\) for addition and \(\displaystyle (a, b) \cdot (c, d) = (ac - bd, ad + bc)\) for multiplication.

Those are the only ones I can think of that would relate to your post.

-Dan

 

[HOI]

I need to explain to a student the meaning of number systems.
I try to explain him that it is like a family of numbers that have some proprity.
If one have an idea to expalin him the term, it will be helpful.
I remind note about the term:
A number system can be complex system number, rational system number, irrational system number ...
And more question I get from him is:
How to calculate the execrise:
"(3, 2) * (3, 2) = ?".
How I need to explain it him.
Ideas will be blessed.

The first thing you will have to explain to him is what "(3, 2)" means!

None of the usual representations of "complex numbers" or "rational numbers" involve pairs. It is possible to define each in terms of pairs of (simpler) numbers but then you have different definitions of "*". For example, the complex number a+ bi can be represented as the pair of real numbers, (a, b). In that case multiplication is defined by (a, b)*(c, d)= (ac- bd, bc+ ad).

In the rational number system (not "rational system number") we can define a rational number as an "equivalence class" of pairs of integers, [(a, b)], where (a, b) is equivalent to (c, d) if and only if ad= bc. In that case the product of the rational number corresponding to the equivalence class containing the pair (a, b) and the rational number corresponding to the equivalence class containing the pair (c, d) is the equivalence class containing the pair (ac, bd).

In the integers, we can define an integer as the equivalence class of pairs of "counting numbers", [(a, b)] where (a, b) is equivalent to (c, d) if and only if a+ d= b+ c. In that case, the product of the integer corresponding to the equivalence class containing the pair (a, b) and the integer corresponding to the equivalence class containing the pair (c, d) is the equivalence class containing the pair (ac+ bd, bc+ad).

So, which pairs are you talking about? What does (a, b) mean?

(There is no "irrational number system". There are, of course, "irrational numbers" but they do not form a "number system" because they are not closed under the usual arithmetic operations of addition and multiplication.)

 

[roni1]

Is in the notation of (a, b) a and b (or a or b) be a complex number for themsleves?
One of my students want to explaio him what is "number system"? Any ideas. (I mean in simple English).
And other question that I want to bring to the class is: "What are complex number?". What in the answers of the student I should notice on?

 

[HOI]

What level class is this? Do you not know what a "number system" is, yourself? You say "I try to explain him that it is like a family of numbers that have some property." I would add that a "number system" also has the basic "arithmetic operations" defined. There is not much point in defining the rational numbers or complex numbers if you can not add or multiply them!

That was why I said earlier that the "irrational numbers" do not form a "number system". Certainly there exist irrational numbers but the addition and multiplication defined on them do not generally stay in the irrational number: \(\displaystyle \sqrt{2}\) and \(\displaystyle -\sqrt{2}\) are irrational numbers but \(\displaystyle \sqrt{2}*\sqrt{2}= 2\) and \(\displaystyle \sqrt{2}+ (-\sqrt{2})= 0\) are not.

Complex numbers are the real numbers together with a new number, i, added to it. Since we want to be able to multiply these numbers, for any real number, b, we have to define its product with this new number "i". What we do is define it to be simply "bi". We want to be able to add real numbers, say the number a, and we write that as "a+ bi". Of course, we want to be able to multiply i with itself and, for historic reasons, the reasons we need to define the complex numbers, we define the product with itself to be -1: \(\displaystyle i*i= i^2= -1\). To define multiplication of complex numbers in general, we use regular "multiplication of binomials": (a+ bi)(c+ di)= a(c+ di)+ bi(c+ di)= ac+ adi+ bci+ bdi^2= ac- bd+ (ad+ bc)i.

Some people object to just 'introducing' this new "symbol", i, when there is no (real) number whose square is -1. They (and I am one of them) prefer to introduce the complex numbers as pairs of numbers, (a, b) with addition and multiplication defined by (a, b)*(c, d)= (ac- bd, ad+ bc). That is exactly the same as identifying (a, b) with a+ bi.

Again, to have a "number system" it is not enough to define the "numbers" themselves only. You also need to define the basic operations so addition and multiplication of the "numbers".