Introduction to axioms of real numbers:
The two distinct real numbers are combined by means of addition and multiplication. These real numbers are must satisfy the following axioms. Many number of axioms are followed by the real numbers. In this article we are going to see about axioms of real numbers with some examples and practice problems.
Types of Axioms
The following Laws are axioms of real number
Associative Addition Law : a + (b + c) = (a + b) + c
Associative Multiplication Law : a * (b * c) = (a * b) * c
Additive identity : a+0 = 0+a = a
Multiplicative identity : a *1= 1*a=a
Commutative Addition Law : a+b = b+a
Commutative Multiplication Law : a * b = b * a
Inverse Law : a + (-a) = 0
Distributive Law : a * (b + c) = a * b + a * c
Examples for Axioms of Real Numbers:
Let take three real numbers 2 and 4 and 6
Problem1:
Apply the real numbers 2 and 4 and 6 in associative addition and Multiplication Laws
Solution:
As per Associative Addition Law: a + ( b + c ) = ( a + b) + c
Here a = 2 and b = 4 and c = 6
2+ (4+6) = (2+4)+6
2+(10) = (6)+6
12 =12 so these real numbers satisfy the associative addition law.
As per Associative Multiplication Law : a * (b * c) = (a * b) * c
2*(4*6) = (2*4)*6
2*(24) = (8)*6
48 = 48 so these real numbers satisfy the Associative Multiplication Law
Problem 2:
Apply the real number 2 in Additive identity and Multiplicative identity
Solution:
As per Additive identity : a+0 = 0+a = a
a = 2
2+0 = 0+2 = 2 So the Additive identity proved.
As per Multiplicative identity : a *1= 1*a=a
a = 2
2*1=1*2 = 2 hence proved.
Problem 3:
Apply the real numbers 2 and 4 in commutative Addition and Multiplication Laws
Solution:
As per commutative Addition Law : a+b = b+a
a = 2 and b=4
2+4 = 4+2
6 = 6
As per commutative Multiplication Law : a * b = b * a
2*4 = 4*2
8 = 8 hence proved.
Between, if you have problem on these topics Find the Average Rate of Change, please browse expert math related websites for more help on who invented calculus.
Practice Problems for Axioms of Real Numbers:
Problem 1:
Apply the real numbers 8 and 5 and 9 in Distributive Law
Ans:112
Problem 2:
Apply the real number 9 in Inverse Law
Ans:0
The two distinct real numbers are combined by means of addition and multiplication. These real numbers are must satisfy the following axioms. Many number of axioms are followed by the real numbers. In this article we are going to see about axioms of real numbers with some examples and practice problems.
Types of Axioms
The following Laws are axioms of real number
Associative Addition Law : a + (b + c) = (a + b) + c
Associative Multiplication Law : a * (b * c) = (a * b) * c
Additive identity : a+0 = 0+a = a
Multiplicative identity : a *1= 1*a=a
Commutative Addition Law : a+b = b+a
Commutative Multiplication Law : a * b = b * a
Inverse Law : a + (-a) = 0
Distributive Law : a * (b + c) = a * b + a * c
Examples for Axioms of Real Numbers:
Let take three real numbers 2 and 4 and 6
Problem1:
Apply the real numbers 2 and 4 and 6 in associative addition and Multiplication Laws
Solution:
As per Associative Addition Law: a + ( b + c ) = ( a + b) + c
Here a = 2 and b = 4 and c = 6
2+ (4+6) = (2+4)+6
2+(10) = (6)+6
12 =12 so these real numbers satisfy the associative addition law.
As per Associative Multiplication Law : a * (b * c) = (a * b) * c
2*(4*6) = (2*4)*6
2*(24) = (8)*6
48 = 48 so these real numbers satisfy the Associative Multiplication Law
Problem 2:
Apply the real number 2 in Additive identity and Multiplicative identity
Solution:
As per Additive identity : a+0 = 0+a = a
a = 2
2+0 = 0+2 = 2 So the Additive identity proved.
As per Multiplicative identity : a *1= 1*a=a
a = 2
2*1=1*2 = 2 hence proved.
Problem 3:
Apply the real numbers 2 and 4 in commutative Addition and Multiplication Laws
Solution:
As per commutative Addition Law : a+b = b+a
a = 2 and b=4
2+4 = 4+2
6 = 6
As per commutative Multiplication Law : a * b = b * a
2*4 = 4*2
8 = 8 hence proved.
Between, if you have problem on these topics Find the Average Rate of Change, please browse expert math related websites for more help on who invented calculus.
Practice Problems for Axioms of Real Numbers:
Problem 1:
Apply the real numbers 8 and 5 and 9 in Distributive Law
Ans:112
Problem 2:
Apply the real number 9 in Inverse Law
Ans:0
