Definition 2.1.1.
The set of natural numbers is denoted by \(\mathbf{N}\) and is described as \(\mathbf{N} = \left\{ \ 1\ ,\ 2,\ 3,\ \ldots\ \right\}\)
Section 2.1 The real number System
Subsection 2.1.1 The set of natural numbers
The history of numbers indicated that the first set of numbers used by the ancient human beings for counting purpose was the set of natural (counting) numbers.
Subsubsection 2.1.1.1 Operations on the set of natural numbers
i) Addition (+).
If two natural numbers \(a\) & \(b\) are added using the operation “+”, then the sum \(a+b\) is also a natural number. If the sum of the two natural numbers \(a\) & \(b\) is denoted by \(c\text{,}\) then we can write the operation as: \(c=a+b\text{,}\) where \(c\) is called the sum and \(a\) & \(b\) are called terms.
Example 2.1.2.
ii) Multiplication \((\times)\).
If two natural numbers \(a\) & \(b\) are multiplied using the operation “\(\times\)”, then the product \(a \times b\) is also a natural number. If the product of the two natural numbers \(a\) & \(b\) is denoted by \(c\text{,}\) then we can write the operation as: \(c = a \times b\text{,}\) where \(c\) is called the product and \(a\) & \(b\) are called factors.
Example 2.1.3.
Properties of addition and multiplication on the set of natural numbers.
i. For any two natural numbers \(a\) & \(b\text{,}\) the sum \(a+b\) is also a natural number. For instance in the above example, \(3\) and \(8\) are natural numbers, their sum \(11\) is also a natural number. In general, we say that the set of natural numbers is closed under addition.
Example 2.1.4.
\(3+8 = 8+3 = 11\text{.}\) In general, we say that addition is commutative on the set of natural numbers.
iii. For any three natural numbers \(a\text{,}\) \(b\) & \(c\text{,}\) \((a+b)+c = a+(b+c)\text{.}\)
Example 2.1.5.
\((3+8)+6 = 3+(8+6) = 17\text{.}\) In general, we say that addition is associative on the set of natural numbers.
iv. For any two natural numbers \(a\) & \(b\text{,}\) the product \(a \times b\) is also a natural number. For instance in the above example, \(3\) and \(4\) are natural numbers, their product \(12\) is also a natural number. In general, we say that the set of natural numbers is closed under multiplication.
Example 2.1.6.
\(3 \times 4 = 4 \times 3 = 12. \) In general, we say that multiplication is commutative on the set of natural numbers.
vi. For any three natural numbers \(a\text{,}\) \(b\) & \(c\text{,}\) \((a \times b) \times c = a \times (b \times c)\text{.}\)
Example 2.1.7.
\((2 \times 4)\times 5 = 2 \times (4 \times 5) = 40\text{.}\) In general, we say that multiplication is associative on the set of natural numbers.
Example 2.1.8.
\(6 \times 1 = 1 \times 6 = 6.\) In general, we say that multiplication has an identity element on the set of natural numbers and \(1\) is the identity element.
For any three natural numbers \(a\text{,}\) \(b\) & \(c\text{,}\) \(a \times (b+c) = (a \times b)+(a \times c)\text{.}\)
Example 2.1.9.
\(3\times (5+7) = (3 \times 5)+ (3 \times 7) = 36\text{.}\) In general, we say that multiplication is distributive over addition on the set of natural numbers.
Note 2.1.10.
Consider two numbers \(a\) and \(b\text{,}\) we say \(a\) is greater than \(b\) denoted by \(a\) > \(b\) if \(a - b\) is positive.
Subsubsection 2.1.1.2 Order Relation in \(\mathbf{N}\)
-
Transitive property:For any three natural numbers \(a\text{,}\) \(b\) & \(c\text{,}\) \(\quad a > b \text{ & } b > c\ \Rightarrow a > c\)
-
Addition property:For any three natural numbers \(a\text{,}\) \(b\) & \(c \text{,}\) \(\quad a > b \Rightarrow a + c > b + c\)
-
Multiplication property:For any three natural numbers \(a\text{,}\) \(b\) and \(c\text{,}\) \(\quad a > b \Rightarrow ac > bc \)
-
Law of trichotomyFor any two natural numbers \(a\) & \(b\) we have \(\quad a > b \text{ or }\ a < b \text{ or } a = b\text{.}\)
Subsubsection 2.1.1.3 Factors of a number
Definition 2.1.11.
If \(a,\ b,\ c\ \in \mathbf{N}\) such that \(ab = c\text{,}\) then \(a\) & \(b\) are factors (divisors) of \(c\) and \(c\) is called product (multiple) of \(a\) & \(b\text{.}\)
Example 2.1.12.
Find the factors of \(15\text{.}\)
Definition 2.1.13.
A number \(a \in \mathbf{N}\) is said to be
-
Even if it is divisible by \(2\text{.}\)
-
Odd if it is not divisible by \(2\text{.}\)
-
Prime if it has only two factors (\(1\) and itself).
-
Composite: if it has three or more factors.
Example 2.1.14.
\(2, 4, 6, . . .\) are even numbers
Example 2.1.15.
\(1, 3, 5, . . .\) are odd numbers
Example 2.1.16.
\(2, 3, 5, . . .\) are prime numbers
Example 2.1.17.
\(4, 6, 8, 9, . . .\) are composite numbers
Remark 2.1.18.
\(1\) is neither prime nor composite.
Subsubsection 2.1.1.4 Prime Factorization
Definition 2.1.19.
Prime factorization of a composite number is the product of all its prime factors.
Example 2.1.20.
-
\(\displaystyle 6 = 2 \times 3\)
-
\(\displaystyle \text{30} = 2 \times 3 \times 5\)
-
\(\displaystyle \text{12} = 2 \times 2 \times 3 = 2^{2} \times 3\)
-
\(\displaystyle 8 = 2 \times 2 \times 2 = 2^{3}\)
-
\(\displaystyle \text{180} = 2^{2} \times 3^{2} \times 5\)
Fundamental Theorem of Arithmetic:
|
|---|
Every composite number can be expressed as a product of its prime factors. This factorization is unique except the order of the factors.
|
Subsubsection 2.1.1.5 Greatest Common Factor (GCF)
Definition 2.1.21.
The greatest common factor (GCF) of two numbers \(a\) & \(b\) is denoted by GCF (\(a\text{,}\) \(b\)) and is the greatest number which is a factor of each of the given number.
Note 2.1.22.
Note: If the GCF of two numbers is \(1\text{,}\) then the numbers are called relatively prime.
Example 2.1.23.
Solution.
Therefore, GCF\((24, 60) = 12\text{.}\)
This method of finding the GCF of two or more numbers is usually lengthy and time consuming. Hence an alternative method (Prime factorization method) is provided as below:
Step 1: Find the prime factorization of each of the natural numbers
Step 2: Form the GCF of the given numbers as the product of every factor that appears in each of the prime factorization but take the least number of times it appears.
Example 2.1.24.
Example 2.1.25.
Solution.
Step 1:
\begin{align*}
20 \amp = 2^2 \times 5 \\
80 \amp = 2^4 \times 5 \\
450 \amp= 2 \times 3^2 \times 5^2
\end{align*}
Step 2: The factors that appear in all cases are \(2\) and \(5\text{,}\) but take the numbers with the least number of times.
\(\therefore GCF(20,80,450) = 2 \times 5 = 10\)
Subsubsection 2.1.1.6 Least Common Multiple (LCM)
Definition 2.1.26.
The least common multiple (LCM) of two numbers \(a\) & \(b\) is denoted by LCM (\(a\text{,}\) \(b\)) and is the least number which is a multiple of each of the given number.
Example 2.1.27.
Solution.
Now, \(M_{18} = {18,36,54,72,90,108,126,144, \,...}\) and \(M_{24} = {24,48,72,96,120,144,\, ...}\)
Therefore, LCM \((18, 24) = 72\text{.}\)
This method of finding the LCM of two or more numbers is usually lengthy and time consuming. Hence an alternative method (Prime factorization method) is provided as below:
Step 1: Find the prime factorization of each of the natural numbers
Step 2: Form the LCM of the given numbers as the product of every factor that appears in any of the prime factorization but take the highest number of times it appears.
Example 2.1.28.
Example 2.1.29.
Solution.
Step 1:
\begin{align*}
20 \amp = 2^2 \times 5 \\
80 \amp = 2^4 \times 5 \\
450 \amp= 2 \times 3^2 \times 5^2
\end{align*}
Step 2: The factors that appear in any cases are \(2 , 3\) and \(5\text{,}\) but take the numbers with the highest number of times.
\(\therefore GCF (20,80,450) = 2^4 \times 3^2 \times 5^2 = 3600\)
Subsubsection 2.1.1.7 Well ordering Principle in the set of natural numbers
Proposition 2.1.30.
Every non-empty subset of the set of natural numbers has smallest (least) element.
Example 2.1.31.
\(\ A = \left\{ \ 2\ ,\ 3,\ 4,\ \cdots\ \right\}\ \subseteq \ N\text{.}\ \text{smallest}\ \text{element}\ \text{of}\ A = 2\text{.}\)
Note 2.1.32.
The set of counting numbers including zero is called the set of whole numbers and is denoted by W. i.e \(W =\left\{ \ 0,\ 1\ ,\ 2,\ 3,\ \ldots\ \right\}\)
Subsubsection 2.1.1.8 Principle of Mathematical Induction
Mathematical induction is one of the most important techniques used to prove in mathematics. It is used to check conjectures about the outcome of processes that occur repeatedly according to definite patterns. We will introduce the technique with examples.
The method is used to prove different propositions involving positive integers using three steps:
Example 2.1.33.
Show that \(1 + 3 + 5 + \cdots + (2n - 1) = n^{2}\text{.}\)
Solution.
Proof.
Step1. For \(n=1, 1=1^2\) which is true.
Step2. Assume that it is true for \(n=k\)
Step3. We should show that it is true for \(n=k+1.\)
\begin{align*}
\text{Now } \,\, 1+3+5+\cdots+(2k-1)+(2k+1) \amp = k^2 + (2k+1) \\
\amp = k^2 + 2k + 1\\
\amp = (k+1)^2 \quad \text{ which is the required result.}
\end{align*}
Example 2.1.34.
Show that \(1 + 2 + 3 + \cdots + (n) = \frac{n (n + 1)}{2}\text{.}\)
Solution.
Proof.
Step1. For \(n=1, 1=\frac{1(1+1)}{2}\) which is true.
Step2. Assume that it is true for \(n=k\)
Step3. We should show that it is true for \(n=k+1\)
\begin{align*}
\text{Now } \, 1+2+3+\cdots+(k)+(k+1) \amp = \frac{k(k+1)}{2} + (k+1)\\
\amp = \frac{k(k+1)+2(k+1)}{2}\\
\amp = \frac{(k+1)(k+2)}{2} \,\, \text{which is the required result.}
\end{align*}
Example 2.1.35.
Show that \(5^{n} + 6^{n} < 9^{n}\ \text{for}\ n \geq 2\text{.} \)
Solution.
Proof.
Step2. Assume that it is true for \(n=k\text{.}\)
i.e. \(5^k + 6^k < 9^k\text{.}\)
Step3. We should show that it is true for \(n=k+1\)
Claim: \(5^{k+1} + 6^{k+1} < 9^{k+1}\text{.}\)
Now \(5^{k+1} + 6^{k+1} = 5.5^k + 6.6^k < 6.5^k + 6.6^k\)
\(\qquad \qquad \qquad \qquad = 6(5^k + 6^k)\)
\(\qquad \qquad \qquad \qquad < 9(5^k + 6^k)\)
\(\qquad \qquad \qquad \qquad < 9(9^k) = 9^{k+1}\)
\(\Rightarrow 5^{k+1} + 6^{k+1} < 9^{k+1} \text{ which is the required format.}\)
Checkpoint 2.1.36.
1
docs.google.com/forms/d/e/1FAIpQLSfSNI6CXkmgeSZJh6v0WKkeD9MJ9g4pEQ9r0JaowD4ovNxj5w/viewform?usp=pp_url&entry.699375810=questions%2Ftop%2FMathematics-for-NS-%26-SS-23-24-Question-Bank%2FThe-Real-and-Complex-Number-Systems-%28NS%29%2FThe-set-of-natural-numbers%2FFinding-the-LCM-and-GCF-of-three-integers.xml&entry.2077830997=source%2Fthe-real-and-complex-number-systems%2Fsections%2Fsubsections%2Fsec-the-real-number-system%2Fsubsec-the-set-of-natural-numbers.ptx
Checkpoint 2.1.37.
2
docs.google.com/forms/d/e/1FAIpQLSfSNI6CXkmgeSZJh6v0WKkeD9MJ9g4pEQ9r0JaowD4ovNxj5w/viewform?usp=pp_url&entry.699375810=questions%2Ftop%2FMathematics-for-NS-%26-SS-23-24-Question-Bank%2FThe-Real-and-Complex-Number-Systems-%28NS%29%2FThe-set-of-natural-numbers%2FProving-the-sum-of-sequence-of-natural-numbers-using-mathematical-induction.xml&entry.2077830997=source%2Fthe-real-and-complex-number-systems%2Fsections%2Fsubsections%2Fsec-the-real-number-system%2Fsubsec-the-set-of-natural-numbers.ptx
Checkpoint 2.1.38.
3
docs.google.com/forms/d/e/1FAIpQLSfSNI6CXkmgeSZJh6v0WKkeD9MJ9g4pEQ9r0JaowD4ovNxj5w/viewform?usp=pp_url&entry.699375810=questions%2Ftop%2FMathematics-for-NS-%26-SS-23-24-Question-Bank%2FThe-Real-and-Complex-Number-Systems-%28NS%29%2FThe-set-of-natural-numbers%2FProving-the-set-of-natural-numbers-using-mathematical-induction.xml&entry.2077830997=source%2Fthe-real-and-complex-number-systems%2Fsections%2Fsubsections%2Fsec-the-real-number-system%2Fsubsec-the-set-of-natural-numbers.ptxSubsection 2.1.2 The set of Integers
As the knowledge and interest of human beings increased, it was important and obligatory to extend the natural number system. For instance to solve the equation \(x+1= 0\text{,}\) the set of natural numbers was not sufficient. Hence the set of integers was developed to satisfy such extended demands.
Definition 2.1.39.
The set of integers is denoted by Z and described as \(\mathbf{Z} = \left\{ \ldots , - 2, - 1, 0, 1 , 2, \ldots \right\}\)
Subsubsection 2.1.2.1 Operations on the set of integers
i) Addition \((+)\).
If two integers \(a\) & \(b\) are added using the operation “\(+\)”, then the sum \(a+b\) is also an integer. If the sum of the two integers \(a\) & \(b\) is denoted by \(c\text{,}\) then we can write the operation as: \(c=a+b\text{,}\) where \(c\) is called the sum and \(a\) & \(b\) are called terms.
Example 2.1.40.
ii) Subtraction \((-)\).
For any two integers \(a\) & \(b\text{,}\) the operation of subtracting \(b\) from \(a\text{,}\) denoted by \(a - b\) is defined by \(a - b = a + ( - b)\text{.}\) This means that subtracting \(b\) from \(a\) is equivalent to adding the additive inverse of \(b\) to \(a\text{.}\)
Example 2.1.41.
\(7 - 5 = 7 + ( - 5) = 2\)
iii) Multiplication (\(\times\)).
If two integers \(a\) & \(b\) are multiplied using the operation “\(\times\)”, then the product \(a\)\(\times\)\(b\) is also an integer. If the product of the two integers \(a\) & \(b\) is denoted by \(c\text{,}\) then we can write the operation as: \(c = a \times b\text{,}\) where \(c\) is called the product and \(a\) & \(b\) are called factors.
Example 2.1.42.
Properties of addition and multiplication on the set of integers.
i) For any two integers \(a\) & \(b\text{,}\) the sum \(a+b\) is also an integer. For instance in the above example, \(4\) and \(9\) are integers, their sum \(13\) is also an integer. In general, we say that the set of integers is closed under addition.
Example 2.1.43.
\(4+9 = 9+4 = 13\text{.}\) In general, we say that addition is commutative on the set of integers.
Example 2.1.44.
\((5+9)+8 = 5+(9+8) = 22\text{.}\) In general, we say that addition is associative on the set of integers.
Example 2.1.45.
\(7+0 = 0+7 = 7\text{.}\) In general, we say that addition has an identity element on the set of integers and \(0\) is the identity element.
Example 2.1.46.
\(4+-4 = -4+4 = 0\text{.}\) In general, we say that every integer \(a\) has an additive inverse denoted by\(- a\text{.}\)
vi) For any two integers \(a\) & \(b\text{,}\) the product \(a \times b\) is also an integer. For instance in the above example, \(4\) and \(7\) are integers, their product \(28\) is also an integer. In general, we say that the set of integers is closed under multiplication.
Example 2.1.47.
\(4 \times 7 = 7\times4 = 28\text{.}\) In general, we say that multiplication is commutative on the set of integers.
viii) For any three integers \(a\text{,}\) \(b\) & \(c\text{,}\) \((a \times b) \times c = a \times (b \times c)\text{.}\)
Example 2.1.48.
\((3 \times 5) \times 4 = 3 \times (5 \times 4) = 60 \text{.}\) In general, we say that multiplication is associative on the set of integers.
Example 2.1.49.
\(5 \times 1 = 1 \times 5 = 5 \text{.}\) In general, we say that multiplication has an identity element on the set of integers and \(1\) is the identity element.
x) For any three integers \(a\text{,}\) \(b\) & \(c\text{,}\) \(a \times (b+c) = (a \times b)+(a \times c)\text{.}\)
Example 2.1.50.
\(4 \times (5+6) = (4 \times 5)+ (4 \times 6) = 44\text{.}\) In general, we say that multiplication is distributive over addition on the set of integers.
Subsubsection 2.1.2.2 Order Relation in Z
-
Transitive property: For any three integers \(a\text{,}\) \(b\) & \(c\text{,}\) \(a > b\ \) & \(\ b > c\ \Rightarrow a > c\)
-
Addition property: For any three integers \(a\text{,}\) \(b\) & \(c\text{,}\) \(a > b\ \Rightarrow \ a + c > b + c\)
-
Multiplication property: For any three integers \(a\text{,}\) \(b\) and \(c\text{,}\) where \(c\) > \(0\text{,}\) \(a > b \Rightarrow \text{ ac } > \text{ bc }\)
-
Law of trichotomy: For any two integers \(a\) & \(b\) we have \(a > b \text{ or } a < b \text{ or } a = b\text{.}\)
Exercises Exercises
1.
2.
There are between \(50\) and \(60\) number of eggs in a basket. When Loza counts by 3’s, there are \(2\) eggs left over. When she counts by 5’s, there are \(4\) left over. How many eggs are there in the basket?
3.
The GCF of two numbers is \(3\) and their LCM is \(180\text{.}\) If one of the numbers is \(45\text{,}\) then find the second number.
4.
Using Mathematical Induction, prove the following:
-
\(\displaystyle 2 + 4 + 6 + \cdots + 2n = n(n + 1)\)
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\(\displaystyle 1^{2} + 2^{2} + 3^{2} + \cdots + n^{2} = \frac{n (n + 1)(2n + 1)}{6}\)
-
\(\displaystyle 1^{3} + 2^{3} + 3^{3} + \cdots + n^{3} = \frac{n^{2}\ (n + 1)^{2}}{4}\)
-
\(\displaystyle \frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + \cdots + \frac{1}{n(n + 1)} = \frac{n}{n + 1}\)
Checkpoint 2.1.51.
4
docs.google.com/forms/d/e/1FAIpQLSfSNI6CXkmgeSZJh6v0WKkeD9MJ9g4pEQ9r0JaowD4ovNxj5w/viewform?usp=pp_url&entry.699375810=questions%2Ftop%2FMathematics-for-NS-%26-SS-23-24-Question-Bank%2FThe-Real-and-Complex-Number-Systems-%28NS%29%2FThe-set-of-integers%2Fproving-Use-the-principle-of-mathematical-induction-to-show-that-2n.xml&entry.2077830997=source%2Fthe-real-and-complex-number-systems%2Fsections%2Fsubsections%2Fsec-the-real-number-system%2Fsubsec-the-set-of-integers.ptx
Checkpoint 2.1.52.
5
docs.google.com/forms/d/e/1FAIpQLSfSNI6CXkmgeSZJh6v0WKkeD9MJ9g4pEQ9r0JaowD4ovNxj5w/viewform?usp=pp_url&entry.699375810=questions%2Ftop%2FMathematics-for-NS-%26-SS-23-24-Question-Bank%2FThe-Real-and-Complex-Number-Systems-%28NS%29%2FThe-set-of-integers%2Fproving-the-set-of-integers-using-disproving-by-counterexamples-.xml&entry.2077830997=source%2Fthe-real-and-complex-number-systems%2Fsections%2Fsubsections%2Fsec-the-real-number-system%2Fsubsec-the-set-of-integers.ptxSubsection 2.1.3 The set of rational numbers
As the knowledge and interest of human beings increased with time, it was again necessary to extend the set of integers. For instance to solve the equation \(2x +1 = 0\text{,}\) the set of integers was not sufficient. Hence the set of rational numbers was developed to satisfy such extended needs.
Definition 2.1.53.
Any number that can be expressed in the form \(\frac{a}{b}\text{,}\) where \(a\) and \(b\) are integers and \(b \neq 0\text{,}\) is called a rational number. The set of rational numbers denoted by \(Q\) is described by
\begin{equation*}
Q = \left\{ \frac{a}{b}: a \text{ and } b \text{ are integers and } b \neq 0 \right\}
\end{equation*}
Subsubsection 2.1.3.1 Operations on the set of rational numbers
i) Addition \((+)\).
If two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) are added using the operation “+”, then the sum defined as \(\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\) is also a rational number.
Example 2.1.54.
\(\frac{1}{2} + \frac{3}{5} = \frac{11}{10}\)
ii) Subtraction \((-)\).
For any two rational numbers \(\frac{a}{b}\) & \(\frac{c}{d}\text{,}\) the operation of subtracting \(\frac{c}{d}\) from \(\frac{a}{b}\text{,}\) denoted by \(\frac{a}{b} - \frac{c}{d}\) is defined by \(\frac{a}{b} - \frac{c}{d} = \frac{a}{b} +(-\frac{c}{d})\text{.}\)
Example 2.1.55.
\(\frac{1}{2} - \frac{3}{5} = \frac{- 1}{10}\)
iii) Multiplication \((\times)\).
If two rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) are multiplied using the operation “\(\times \)”, then the product defined as \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\) is also a rational number.
Example 2.1.56.
\(\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}\)
iv) Division \((\div)\).
For any two rational numbers \(\frac{a}{b}\) & \(\frac{c}{d}\text{,}\) dividing \(\frac{a}{b}\) by \(\frac{c}{d}\) is defined by
\(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}, \, \, c \neq 0.\)
Example 2.1.57.
\(\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}\)
Properties of addition and multiplication on the set of rational numbers.
-
The set of rational numbers is closed under addition and multiplication.
-
Addition and multiplication are both commutative on the set of rational numbers.
-
Addition and multiplication are both associative on the set of rational numbers.
-
\(0\) is the additive identityi.e., \(\frac{a}{b} + 0 = 0 + \frac{a}{b} = \frac{a}{b}\text{.}\)
-
Every rational number has an additive inverse.i.e., \(\frac{a}{b} + ( - \frac{a}{b}) = -\frac{a}{b} + \frac{a}{b} = 0\text{.}\)
-
\(1\) is the multiplicative identityi.e., \(\frac{a}{b} \times 1 = 1 \times \frac{a}{b} = \frac{a}{b}\text{.}\)
-
Every non-zero rational number has a multiplicative inverse.i.e., \(\frac{a}{b} \times \frac{b}{a} = \frac{b}{a} \times \frac{a}{b} = 1\text{.}\)
Subsubsection 2.1.3.2 Order Relation in Q
-
\(\textbf{ Transitive property }\)For any three rational numbers\(\frac{a}{b}\text{,}\) \(\frac{c}{d}\) & \(\frac{e}{f}\) \(\frac{a}{b} > \frac{c}{d} \& \frac{c}{d} > \frac{e}{f}\ \Rightarrow \frac{a}{b} > \frac{e}{f}.\)
-
\(\textbf{ Addition property }\)For any three rational numbers \(\frac{a}{b}\text{,}\) \(\frac{c}{d}\) & \(\frac{e}{f}\) \(\frac{a}{b} > \frac{c}{d} \Rightarrow \frac{a}{b} + \frac{e}{f} > \frac{c}{d} + \frac{e}{f}\text{.}\)
-
\(\textbf{ Multiplication property }\)For any three rational numbers \(\frac{a}{b}\text{,}\) \(\frac{c}{d}\text{,}\) \(\frac{e}{f}\) and \(\frac{e}{f} > 0\text{.}\)
-
\(\textbf{ Law of trichotomy }\)For any two rational numbers \(\frac{a}{b}\) & \(\frac{c}{d}\) we have \(\frac{a}{b} > \frac{c}{d}\ \text{ or } \frac{a}{b} < \frac{c}{d}\ \text{ or } \frac{a}{b} = \frac{c}{d}\text{.}\)
Subsubsection 2.1.3.3 Decimal representation of rational numbers
A rational number \(\frac{a}{b}\) can be written in decimal form using long division.
Terminating decimals.
Example 2.1.58.
Express the fraction number \(\frac{25}{4}\) in decimal form.
Non-terminating periodic decimals.
Example 2.1.59.
Express the fraction number \(\frac{25}{3}\) in decimal form.
Now we will see how to convert decimal numbers in to their fraction forms. In earlier mathematics topics, we have seen that multiplying a decimal by \(10\) pushes the decimal point to the right by one position and in general, multiplying a decimal by \(10^n\) pushes the decimal point to the right by \(n\) positions. We will use this fact for the succeeding topics.
Subsubsection 2.1.3.4 Fraction form of decimal numbers
A rational number which is written in decimal form can be converted to a fraction form as \(\frac{a}{b}\) in lowest (simplified) form, where \(a\) and \(b\) are relatively prime.
Terminating decimals.
Consider any terminating decimal number \(n\text{.}\) Suppose \(d\) terminates \(n\) digits after the decimal point. \(d\) can be converted to its fraction form as below:
\(d = d \times 1 = d \times \frac{1}{1} = d \times \left( \frac{10^n}{10^n}\right)\text{.}\)
Example 2.1.60.
Convert the terminating decimal \(3.47\) to fraction form.
Non-terminating periodic decimals.
Consider any non-terminating periodic decimal number \(d\text{.}\) Suppose \(d\) has \(k\) non-terminating digits and \(p\) terminating digits after the decimal point. \(d\) can be converted to its fraction form as below:
\(d = d \times 1 = d \times \frac{1}{1} = d \times \left( \frac{10^{k+p} - 10^k}{10^{k+p} - 10^k}\right)\)
Example 2.1.61.
Convert the non-terminating periodic decimal \(42.5\overline{\text{38}}\) to fraction form.
Solution.
Solution: \(k = 1, p = 2\text{.}\)
\begin{align*}
\therefore d = d \times 1 = d \times \frac{1}{1} \amp = d \times
\left( \frac{10^{k+p} - 10^k}{10^{k+p} - 10^k}\right)\\
\amp = 42.5\overline{\text{38}} \times
\left( \frac{10^3 - 10}{10^3 - 10}\right) \\
\amp= \frac{42538.\overline{\text{38}} - 425.
\overline{\text{38}}}{100 - 10} = \frac{42113}{990}.
\end{align*}
Note 2.1.62.
From the above two cases, we can conclude that both terminating decimals and non-terminating periodic decimals are rational numbers. (Why? Justify).
Subsubsection 2.1.3.5 Non-terminating and non-periodic decimals
Some decimal numbers are neither terminating nor non-terminating periodic. Such types of numbers are called irrational numbers.
Example 2.1.63.
\(62.757757775\ldots\)
Example 2.1.64.
Show that \(\sqrt{2}\) is an irrational number.
Solution.
Proof.
\(\begin{matrix}
\ \text{Suppose}\ \sqrt{2}\ \text{is}\ a\ \text{rational}\ \text{number} \\
\ \Rightarrow \sqrt{2} = \frac{a}{b},\ \text{where}\ \text{GCF}(a,\ b) = 1 \\
\ \Rightarrow 2 = \frac{a^{2}}{b^{2}} \\
\ \Rightarrow a^{2} = 2b^{2}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}( \ast ) \\
\ \Rightarrow a^{2}\ \text{is}\ \text{even} \\
\ \Rightarrow a\ \text{is}\ \text{even} \\
\ \Rightarrow a\ = 2n\ \text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}(\text{**}) \\
\end{matrix}\)
\(\begin{matrix}
\ \text{Putting}\ \text{this}\ \text{in}\ ( \ast )\ \text{we}\ \text{get}:\ \\
\Rightarrow 4n^{2} = 2b^{2} \\
\Rightarrow b^{2} = 2n^{2} \\
\Rightarrow b^{2}\ \text{is}\ \text{even} \\
\Rightarrow b\ \text{is}\ \text{even} \\
\Rightarrow b = 2m\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}\text{.}(\text{***}) \\
\end{matrix}\)
From (**) and (***) we get a contradiction that GCF (\(a\text{,}\) \(b\)) \(= 1\) which implies that \(\sqrt{2}\) is not a rational number.
Therefore, \(\sqrt{2}\) is an irrational number.
Checkpoint 2.1.65.
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Checkpoint 2.1.66.
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docs.google.com/forms/d/e/1FAIpQLSfSNI6CXkmgeSZJh6v0WKkeD9MJ9g4pEQ9r0JaowD4ovNxj5w/viewform?usp=pp_url&entry.699375810=questions%2Ftop%2FMathematics-for-NS-%26-SS-23-24-Question-Bank%2FThe-Real-and-Complex-Number-Systems-%28NS%29%2FThe-set-of-rational-numbers%2Fproving-by-contradiction-of-an-irrational-number.xml&entry.2077830997=source%2Fthe-real-and-complex-number-systems%2Fsections%2Fsubsections%2Fsec-the-real-number-system%2Fsubsec-the-set-of-rational-numbers.ptxSubsection 2.1.4 The set of real numbers
Definition 2.1.67.
A number is called a real number if and only if it is either a rational number or an irrational number.
The set of real numbers denoted by \(\Re\) can be described as the union of the set of rational and irrational numbers. i.e \(\Re = \) {\(x : x\) is a rational number or an irrational number}.
There is a \(1-1\) correspondence between the set of real numbers and the number line (For each point in the number line, there is a corresponding real number and vice-versa).
Subsubsection 2.1.4.1 Operations on the set of real numbers
i) Addition (\(+\)).
If two real numbers are added using the operation “\(+\)”, then the sum is also a real number.
ii) Subtraction (\(-\)).
For any two real numbers \(a\) & \(b\text{,}\) the operation of subtracting \(b\) from \(a\text{,}\) denoted by \(a - b\) is defined by \(a - b = a +(-b)\text{.}\)
iii) Multiplication (\(\times\)).
If two real numbers \(a\) and \(b\) are multiplied using the operation “\(\times\)”, then the product defined as \(a \times b = ab\) is also a real number.
iv) Division (\(\div\)).
For any two real numbers \(a\) & \(b\text{,}\) dividing \(a\) by \(b\) is defined by \(a \div b = a \times \frac{1}{b}, b \neq 0\text{.}\)
Properties of addition and multiplication on the set of real numbers.
-
The set of real numbers is closed under addition and multiplication.
-
Addition and multiplication are commutative on the set of real numbers.
-
Addition and multiplication are associative on the set of real numbers.
-
\(0\) is the additive identity.i.e., \(a + 0 = 0 + a = a\text{.}\)
-
Every real number has an additive inverse.i.e., \(a + (-a) = -a + a = 0\)
-
\(1\) is the multiplicative identity.\(a \times 1 = 1 \times a = a\text{.}\)
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Every non-zero real number has a multiplicative inverse.\(a \times \frac{1}{a} = \frac{1}{a} \times a = 1\)
Subsubsection 2.1.4.2 The real number and the number line
One of the most important properties of the real number is that it can be represented graphically by points on a straight line. The point \(0\) is termed as the origin. Points to the right of \(0\) are called positive real numbers and points to the left of \(0\) are called negative real numbers. Each point on the number line corresponds a unique real number and vice-versa.
Geometrically we say \(a\) is greater than \(b\) if \(a\) is located to the right of \(b\) on the number line.
Subsubsection 2.1.4.3 Order Relation in R
-
Transitive property: For any three real numbers\(a\text{,}\) \(b\)& \(c\text{,}\) \(a > b \) & \(\ b > c \Rightarrow a > c \text{.}\)
-
Addition property: For any three real numbers\(a\text{,}\) \(b\)& \(c\text{,}\) \(a > b \Rightarrow a + c > b + c\text{.}\)
-
Multiplication property: For any three real numbers\(a\text{,}\) \(b\text{,}\) \(c\) and \(c > 0\text{,}\) we have \(a > b \Rightarrow ac > bc \text{.}\)
-
Law of trichotomy: For any two real numbers \(a\)& \(b\) we have \(a > b \text{ or } a < b \text{ or } a = b\text{.}\)
Summary of the real number system.
Subsubsection 2.1.4.4 Intervals
Let \(a\) and \(b\) be two real numbers such that \(a < b\text{,}\) then the intervals which are subsets of R with end points \(a\) and \(b\) are denoted and defined as below:
-
\(\lbrack a,\ b\rbrack\ = \left\{ \ x:\ a \leq x \leq b \right\}\) closed interval from \(a\) to \(b\text{.}\)
-
\((a,\ b\rbrack\ = \left\{ \ x:\ a < x \leq b \right\}\) open-closed interval from \(a\) to \(b\text{.}\)
-
\(\lbrack a,\ b)\ = \left\{ \ x:\ a \leq x < b \right\}\) closed-open interval from \(a\) to \(b\text{.}\)
Subsubsection 2.1.4.5 Upper bounds and lower bounds
Definition 2.1.68.
-
A point \(a \in R\) is said to be an upper bound of \(A\) iff \(x \leq a\) for all \(x \in A\text{.}\)
-
An upper bound of \(A\) is said to be least upper bound (lub) iff it is the least of all upper bounds.
-
A point \(a \in R\) is said to be lower bound of \(A\) iff \(x \geq a\) for all \(x \in A\text{.}\)
-
A lower bound of \(A\) is said to be greatest lower bound (glb) iff it is the greatest of all lower bounds.
Example 2.1.69.
Consider the set \(A = \lbrack 2,\ 5) \subseteq \Re\text{.}\)
Example 2.1.70.
Solution.
\(A = \left\{ \ 1,\ \frac{1}{2},\ \frac{1}{3},\ \cdots \right\}\)
-
lower bounds are \(\cdots, -3, -2, 0\)
-
upper bounds are \(1,3,\frac{9}{2},50,\cdots\)
Based on the above definitions, we can define the completeness property of real numbers as below.
Subsubsection 2.1.4.6 Completeness property of real number (R)
Completeness property of real numbers states that: Every non-empty subset of \(\Re\) that has lower bounds has glb and every non-empty subset of \(\Re\) that has upper bounds has a lub.
Exercises Exercises
1.
Express each of the following rational numbers as decimal:
-
\(\displaystyle \frac{4}{9}\)
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\(\displaystyle \frac{3}{25}\)
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\(\displaystyle \frac{11}{7}\)
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\(\displaystyle -5\frac{2}{3}\)
-
\(\displaystyle \frac{2}{77}\)
2.
Write each of the following as decimal and then as a fraction:
3.
Write each of the following in meters as a fraction and then as a decimal
4.
Classify each of the following as terminating or non-terminating periodic
-
\(\displaystyle \frac{5}{13}\)
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\(\displaystyle \frac{7}{10}\)
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\(\displaystyle \frac{69}{64}\)
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\(\displaystyle \frac{11}{60}\)
-
\(\displaystyle \frac{5}{12}\)
5.
Convert the following decimals to fractions:
-
\(\displaystyle 3.2 \overline{5}\)
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\(\displaystyle 0.3 \overline{14}\)
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\(\displaystyle 0. \overline{275}\)
6.
Determine whether the following are rational or irrational:
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\(\displaystyle 2.7 \overline{5}\)
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\(\displaystyle 0.272727\cdots\)
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\(\displaystyle \sqrt{8} - \frac{1}{\sqrt{2}}\)
7.
Which of the following statements are true and which of them are false?
-
The sum of any two rational numbers is rational
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The sum of any two irrational numbers is irrational
-
The product of any two rational numbers is rational
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The product of any two irrational numbers is irrational
8.
Find two rational numbers between \(\frac{1}{3} \text{ and } \frac{1}{2} \text{.}\)
Checkpoint 2.1.71.
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Checkpoint 2.1.72.
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Checkpoint 2.1.73.
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Checkpoint 2.1.74.
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