The test below has 118 basic questions in the number system, including real numbers, rational numbers, irrational numbers, integers, fractions, and decimals.
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Property  Operation  Symbol  Natural Numbers  Whole Numbers  Integers  

Closure (A Sense of Conclusion)  
Addition  a+b = c  If a & b are Natural numbers. c is also a Natural numbers  Closure  2+3 = 5  If a & b are Whole numbers. c is also a Whole numbers  Closure  "2+3 = 5 0+3 = 3" 
If a & b are Integers. c is also a Integer.  Closure  2+3 = 5 0+3 = 3 

Subtraction  ab = c  "If a & b are Natural numbers. c can be Natural numer or an Integers" 
Not Closure  23 = 1  "If a & b are Whole numbers. c can be a Whole numer or an Integer" 
Not Closure  "23 = 1 33 = 0 40 = 4" 
"If a & b are Integers. c is also a Integer." 
Closure  "23 = 1 33 = 0 40 = 4 23 = 5" 

Multiplication  axb = c  "If a & b are Natural numbers. c is also a Natural numbers" 
Closure  2x3 = 6  "If a & b are Whole numbers. c is also a Whole numbers" 
Closure  "2x3 = 6 0x3 = 0" 
"If a & b are Integers. c is also an Integer." 
Closure  "2x3 = 6 0x3 = 0 2x3 = 6 2x3 = 6" 

Division  a/b = c  "If a & b are Natural numbers. c can be a Fraction" 
Not Closure  2÷3 = 2/3  "If a & b are Whole numbers. c can be a Whole number, Fraction or Not defined" 
Not Closure  "2÷3 = 2/3 0÷2 = 0 3÷0 = Not defined" 
"If a & b are Integers. c can be a Integer, Fraction, Rational Number or Not defined" 
Not Closure  "2÷3 = 2/3 2÷3 =  (2/3) 0÷2 = 0 3÷0 = Not defined" 
Property  Operation  Symbol  Result  Example 

Commutative (A relation to an exchange)  
Addition  a+b = b+a  Commutative  2+3 = 3+2 = 5 0+2 = 2+0 = 2 

Subtraction  ab ≠ ba  Not Commutative  23 ≠ 32 20 ≠ 02 

Multiplication  axb = bxa  Commutative  2x3 = 3x2 = 6 0x2 = 2x0 = 0 

Division  a/b ≠ b/a  Not Commutative  2÷3 ≠ 3/2 
Property  Operation  Symbol  Result  Example 

Associative (A cooperative link between numbers)  
Addition  (a+b)+c = a+(b+c)  Associative  (2+3)+5 = 2+(3+5) => 5+5 = 2+8 => 10 = 10" 

Subtraction  (ab)c ≠ a(bc)  Not Associative  (85)2 = 8(52) => 32 = 83 => 1 ≠ 5" 

Multiplication  (axb)xc = ax(bxc)  Associative  (2x3)x5 = 2x(3x5) => 6x5 = 2x15 => 30 = 30" 

Division  (a/b)/c = a/(b/c)  Not Associative  (16÷4)÷2 = 16÷(4÷2) => 4÷2 = 16÷2 => 2 ≠ 8" 
Property  Operation  Symbol  Result  Example 

Distributive (The action of sharing)  
Multiplication  ax(b+c) = (axb)+(axc)  Distributive  2x(3+5) = (2x3)+(2x5) => 6 + 10 => 16 
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