Introduction to Real numbers $\mathbb{R}$
Today i will be giving a brief tutorial on the real numbers and its algebraic structure for MAT206(Real analysis I) course.
For those who are still new to the real number system.
A real number is simply a value that represents a quantity along a line.
The word "real" was introduced by Rene Descartes in the 17th century while distinguishing between real and imaginary roots of polynomials.
We must also note that the real number include all the rational numbers,such as the integer -4 and fraction $\frac{1}{3}$ and all irrational numbers such as $\sqrt{2}=1.414121356...$.we also have in this irrationals the transcendental numbers such as the $\pi=3.142...$.
The real numbers can also be thought of as point along the number line or real line,where the points corresponding to integers are equally spaced.
Now that is an intro to the real number system,and we can say that is just a basic introduction.
But today i am more concern about the algebraic structure of the real number system.
Algebraic Structure of the Reals
In the algebraic structure of the real number system, we are only concerned with some rules or better still properties that make up the real numbers, and this axioms are going to be useful in proving so many discussions.
The algebraic structure is simply divided into two namely:
(1) The Field Structure
(2) The Order Structure
Now lets begin with the field axiom.
(1)The Field structure
The field structure i will be listing here are going to be divided into 2 namely "Additive axiom" and "Multiplicative axiom".
The field axioms are used for performing algebraic manipulations in the real number system.
Additive Axioms
Given any $a,b,c\in{\mathbb{R}}$
(i) the real numbers are cummutative i.e a+b=b+a, meaning if we take any two real numbers say $1$ and $2$ then their addition is cummuttive $1+2=2+1$.
(ii) They are Associative in the sense that $(a+b)+c=a+(b+c)$ is true e.g if we pick three real numbers say $2,5,7$ then $(2+5)+7=2+(5+7)$ is true.
(iii) Existence of unique number $0\in{\mathbb{R}}$,what we mean by a unique number is, a number that has no effect when it is applied in addition. e.g $0+a=a+0=a$, hence we can see 0 has no effect.
(iv) Existence of additive inverse, this means that for any $a\in\mathbb{R}$ then there exists $-a$ such that $a+(-a)=0$
Multiplicative Axioms
for any $a,b,c\in{\mathbb{R}}$ then the following simply holds:
(v) cummuttivity: $ab=ba$ meaning multiplications also is cummutative like the addition axiom.
(vi) Associative: $(ab)c=a(bc)$, the real number is associative under multiplication.
(vii) Existence of identity: There exists a unique or identity element under multiplication, the unique or identity element in multiplicative rule is $1\in\mathbb{R}$ such that
$a1=1a=1$.
(viii) for any number $a\in\mathbb{R}$ there is a corresponding number $a^{-1}$ with the property that $aa^{-1}=a^0=1$. e.g for any $2\in\mathbb{R}$ then there exists $-2$ such that $2\times{2^{-1}}=1$
and lastly
(ix) Distributivity: The real numbers are distributive under multiplication i.e $(a+b)c=ac+bc$
(2) The order Structure
The real numbers exhibits the order structure($<$).below are the order structure of the reals.
for $a,b,c\in\mathbb{R}$ then the following holds:
(i) Tricotomy Law: this law asserts that $a=b$, $a<b$ or $b>a$.This implies that when we say $2<4$ then $4>2$ which is true.
(ii) Transitive Law: This law asserts that if $a<b$ is true and $b<c$ is true then definitely $a<c$ is true.
e.g given the set of numbers $A=\{2,3,4\}$ then we can see that $2<3$, and $3<4$ holds then automatically $2<4$ also holds.
(iii) Monotricity: This law asserts that if $a<b$ is true then $a+c=b+c$ is also true. e.g given the set of numbers $A=\{2,3,4\}$ then we can see that $2<3$, then $2+4<3+4$ holds.(iv) Monotone Law: if $a<b$ is true then $a.c<b.c$ is also true for $c>0$.
And that is the end of the tutorial, in my next tutorial i will be teaching how to evaluate the absolute value.
Thank You.
Today i will be giving a brief tutorial on the real numbers and its algebraic structure for MAT206(Real analysis I) course.
For those who are still new to the real number system.
A real number is simply a value that represents a quantity along a line.
The word "real" was introduced by Rene Descartes in the 17th century while distinguishing between real and imaginary roots of polynomials.
We must also note that the real number include all the rational numbers,such as the integer -4 and fraction $\frac{1}{3}$ and all irrational numbers such as $\sqrt{2}=1.414121356...$.we also have in this irrationals the transcendental numbers such as the $\pi=3.142...$.
The real numbers can also be thought of as point along the number line or real line,where the points corresponding to integers are equally spaced.
Now that is an intro to the real number system,and we can say that is just a basic introduction.
But today i am more concern about the algebraic structure of the real number system.
Algebraic Structure of the Reals
In the algebraic structure of the real number system, we are only concerned with some rules or better still properties that make up the real numbers, and this axioms are going to be useful in proving so many discussions.
The algebraic structure is simply divided into two namely:
(1) The Field Structure
(2) The Order Structure
Now lets begin with the field axiom.
(1)The Field structure
The field structure i will be listing here are going to be divided into 2 namely "Additive axiom" and "Multiplicative axiom".
The field axioms are used for performing algebraic manipulations in the real number system.
Additive Axioms
Given any $a,b,c\in{\mathbb{R}}$
(i) the real numbers are cummutative i.e a+b=b+a, meaning if we take any two real numbers say $1$ and $2$ then their addition is cummuttive $1+2=2+1$.
(ii) They are Associative in the sense that $(a+b)+c=a+(b+c)$ is true e.g if we pick three real numbers say $2,5,7$ then $(2+5)+7=2+(5+7)$ is true.
(iii) Existence of unique number $0\in{\mathbb{R}}$,what we mean by a unique number is, a number that has no effect when it is applied in addition. e.g $0+a=a+0=a$, hence we can see 0 has no effect.
(iv) Existence of additive inverse, this means that for any $a\in\mathbb{R}$ then there exists $-a$ such that $a+(-a)=0$
Multiplicative Axioms
for any $a,b,c\in{\mathbb{R}}$ then the following simply holds:
(v) cummuttivity: $ab=ba$ meaning multiplications also is cummutative like the addition axiom.
(vi) Associative: $(ab)c=a(bc)$, the real number is associative under multiplication.
(vii) Existence of identity: There exists a unique or identity element under multiplication, the unique or identity element in multiplicative rule is $1\in\mathbb{R}$ such that
$a1=1a=1$.
(viii) for any number $a\in\mathbb{R}$ there is a corresponding number $a^{-1}$ with the property that $aa^{-1}=a^0=1$. e.g for any $2\in\mathbb{R}$ then there exists $-2$ such that $2\times{2^{-1}}=1$
and lastly
(ix) Distributivity: The real numbers are distributive under multiplication i.e $(a+b)c=ac+bc$
(2) The order Structure
The real numbers exhibits the order structure($<$).below are the order structure of the reals.
for $a,b,c\in\mathbb{R}$ then the following holds:
(i) Tricotomy Law: this law asserts that $a=b$, $a<b$ or $b>a$.This implies that when we say $2<4$ then $4>2$ which is true.
(ii) Transitive Law: This law asserts that if $a<b$ is true and $b<c$ is true then definitely $a<c$ is true.
e.g given the set of numbers $A=\{2,3,4\}$ then we can see that $2<3$, and $3<4$ holds then automatically $2<4$ also holds.
(iii) Monotricity: This law asserts that if $a<b$ is true then $a+c=b+c$ is also true. e.g given the set of numbers $A=\{2,3,4\}$ then we can see that $2<3$, then $2+4<3+4$ holds.(iv) Monotone Law: if $a<b$ is true then $a.c<b.c$ is also true for $c>0$.
And that is the end of the tutorial, in my next tutorial i will be teaching how to evaluate the absolute value.
Thank You.
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