7 Axioms
By studying this lesson, you will be able to;
² identify five fundamental axioms of mathematics,
² develop geometrical relationships and solve problems involving calculations
using the five fundamental axioms.
Axioms
Statements which are considered to be self-evident and are accepted without proof
are called axioms. In mathematics, axioms are used to explain facts logically,
develop relationships and reach conclusions.
Euclid, who is considered to be the father of geometry lived in Greece around 300
B.C. He introduced certain axioms related to mathematics in his book “Elements”.
Some of them are unique to geometry. Others are common axioms which can be
used in other areas including algebra.
We consider five common axioms in this lesson. They can be summarized as given
below.
1. Quantities which are equal to the same quantity, are equal.
2. Quantities which are obtained by adding equal quantities to equal quantities,
are equal.
3. Quantities which are obtained by subtracting equal quantities from equal
quantities, are equal.
4. Products which are equal quantities multiplied by equal quantities, are equal.
5. Quotients which are equal quantities divided by nonzero equal quantities,
are equal.
By “quantities” we usually mean lengths, areas, volumes, masses, speeds,
magnitudes of angles, etc.
These five axioms are very important because we can derive many results related to
algebra and geometry by using them. Let us study these axioms in detail.
81
For free distribution
, Axiom 1
Quantities which are equal to the same quantity, are equal.
We can write this axiom briefly as given below.
If b = a and c = a, then b = c.
According to this axiom,
‘If Hasith’s age is the same as Kasun’s and Harsha’s age is also the same as Kasun’s,
then Hasith’s age is the same as Harsha’s.’
How Axiom 1 is used to obtain geometrical results is seen in the simple example
given below.
In the quadrilateral ABCD shown below BC = AB and CD = AB .
B
C
D
A
According to the above axiom,
BC = CD.
Example 1
In the triangle ABC, AB = AC and AB = BC If AC = 5 cm then determine the perimeter
of the triangle ABC.
A
B C
Since AC = 5 cm and AC = AB, according to Axiom 1, AB = 5 cm.
Since AB = 5 cm and AB = BC, according to Axiom 1, BC = 5 cm.
The perimeter of the triangle ABC = AC + BC + AB
= 5 cm + 5 cm + 5 cm
= 15 cm
82
For free distribution
By studying this lesson, you will be able to;
² identify five fundamental axioms of mathematics,
² develop geometrical relationships and solve problems involving calculations
using the five fundamental axioms.
Axioms
Statements which are considered to be self-evident and are accepted without proof
are called axioms. In mathematics, axioms are used to explain facts logically,
develop relationships and reach conclusions.
Euclid, who is considered to be the father of geometry lived in Greece around 300
B.C. He introduced certain axioms related to mathematics in his book “Elements”.
Some of them are unique to geometry. Others are common axioms which can be
used in other areas including algebra.
We consider five common axioms in this lesson. They can be summarized as given
below.
1. Quantities which are equal to the same quantity, are equal.
2. Quantities which are obtained by adding equal quantities to equal quantities,
are equal.
3. Quantities which are obtained by subtracting equal quantities from equal
quantities, are equal.
4. Products which are equal quantities multiplied by equal quantities, are equal.
5. Quotients which are equal quantities divided by nonzero equal quantities,
are equal.
By “quantities” we usually mean lengths, areas, volumes, masses, speeds,
magnitudes of angles, etc.
These five axioms are very important because we can derive many results related to
algebra and geometry by using them. Let us study these axioms in detail.
81
For free distribution
, Axiom 1
Quantities which are equal to the same quantity, are equal.
We can write this axiom briefly as given below.
If b = a and c = a, then b = c.
According to this axiom,
‘If Hasith’s age is the same as Kasun’s and Harsha’s age is also the same as Kasun’s,
then Hasith’s age is the same as Harsha’s.’
How Axiom 1 is used to obtain geometrical results is seen in the simple example
given below.
In the quadrilateral ABCD shown below BC = AB and CD = AB .
B
C
D
A
According to the above axiom,
BC = CD.
Example 1
In the triangle ABC, AB = AC and AB = BC If AC = 5 cm then determine the perimeter
of the triangle ABC.
A
B C
Since AC = 5 cm and AC = AB, according to Axiom 1, AB = 5 cm.
Since AB = 5 cm and AB = BC, according to Axiom 1, BC = 5 cm.
The perimeter of the triangle ABC = AC + BC + AB
= 5 cm + 5 cm + 5 cm
= 15 cm
82
For free distribution
