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- Definitions: FSc Part 1 (Mathematics): PTB @fsc-part1-ptb
- ptb:definitions-aurang-zaib]] ===== Chapter 01: Number system ===== * **Rational number:** A number which can be written in the form of $\frac{p}{q}$, where $p,q \in \mathbb{Z}$, $q\neq 0$, is called a rational number * **Irrational number:** A real number which c
- Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib @fsc-part1-ptb
- for his valuable contribution. =====Chapter 01: Number System===== ====Rational Number==== A number which can be expressed in the form \( \dfrac{p}{q} \), where \( p, q \in \mathbb{Z} \) and \( q \neq 0 \), is termed as a rational number. ===Example==== \( \dfrac{3}{4} \), \( \dfrac{7}{
- Question 7 Exercise 6.4 @math-11-kpk:sol:unit06
- . Find the probability of getting doublet of even numbers. ====Solution==== The sample space rolling a pai... 5) & (6,6) \end{array}\right]\end{align} So total number of sample points are $$n(S)=6 \times 6=36$$ doublet of even numbers. Let \begin{align}A&=\{(2,2),(4,4),(6,6)\}\\ n(... nce the possibility of getting doublet of an even number is: $$P(A)=\dfrac{n(A)}{n(S)}=\dfrac{3}{36}=\dfra
- Question 9 Exercise 6.3 @math-11-kpk:sol:unit06
- re $7$ and total women are $6.$ Therefore, Total number of persons $=7+6=13$ Committee consist of 8 pers... e contain exactly four men and four women. Total number of different ways that four men to be selected are: ${ }^7 C_4$. Total number of different ways that four women to be selected ... . By fundamental principle of counting the total number of different committees that will exactly contain
- Question 13 Exercise 6.2 @math-11-kpk:sol:unit06
- war, Pakistan. =====Question 13(i)===== Find the number of permutation of word "Excellence." How many of ... in with $\mathrm{E}$ ? ====Solution==== The total number of letters in 'Excellence' are: $n=10$, out of wh... 2$ are $C$. Therefore, \begin{align}\text{total number of permutations are} &=\left(\begin{array}{c} n ... d $m_3=2$ are $C$. Therefore, \begin{align}\text{Number of permulations are} &=\left(\begin{array}{c} n \
- MTH321: Real Analysis I (Spring 2023) @atiq
- om Chapter 02** - A convergent sequence of real number has one and only one limit (i.e. limit of the seq... }$ also converges to $s$. - For each irrational number $x$, there exists a sequence $\left\{ {{r}_{n}} \right\}$ of distinct rational numbers such that $\underset{n\to \infty }{\mathop{\lim... _{n}}}$ is convergent if and only if for any real number $\varepsilon >0$, there exists a positive integer
- Number Theory: Handwritten Notes @notes
- ====== Number Theory: Handwritten Notes ====== {{ :notes:number-theory-handwritten-notes.jpg?nolink&600|Number Theory: Handwritten Notes}} The study of the characteristi... of the positive integers (1, 2, 3,...) is called number theory. It is significant because it has numerous
- Number Theory by Dr Muhammad Umer Shuaib @notes
- ====== Number Theory by Dr Muhammad Umer Shuaib ====== {{ :notes:number-theory-muzammil-tanveer.jpg?nolink&600|Number Theory Notes}} A subfield of mathematics called number theory studies the characteristics of positive int
- Algebraic Number Theory Notes by Anwar Khan @notes
- ====== Algebraic Number Theory Notes by Anwar Khan ====== {{ :notes:algebraic-number-theory-notes-anwar-khan.jpg?nolink|Algebraic Number Theory Notes by Anwar Khan}} Algebraic number theory is a subfield of number theory that studies integers
- Chapter 01: Complex Numbers @bsc:notes_of_mathematical_method
- ====== Chapter 01: Complex Numbers ====== {{ :bsc:notes_of_mathematical_method:ch01-methods-ads.jpg?nolink&640x800|Chapter 01 Complex Numbers Methods}} Notes of the book Mathematical Method ... Ilmi Kitab Khana, Lahore - PAKISTAN. A complex number is an element $(x,y)$ of the set $$ \mathbb{R}^2=... C}$. ==== Contents and summary ==== * Complex numbers * Properties of complex numbers * The Argand
- Chapter 01: Number System @fsc:fsc_part_1_solutions
- ====== Chapter 01: Number System ====== {{ :fsc:fsc_part_1_solutions:fsc-1-chap-01-ptb.jpg?nolink|Chapter 01: Number System}} Notes (Solutions) of Chapter 01: Number System, Textbook of Algebra and Trigonometry Class XI (M... ahore. ==== Contents & summary ==== * Rational numbers and irrational numbers * Properties of real nu
- Number Theory Notes by Anwar Khan @notes
- ====== Number Theory Notes by Anwar Khan ====== {{ :notes:number-theory-notes-anwar-khan.jpg?nolink|Number Theory Notes by Anwar Khan}} Mathematicians who specialize in number theory examine the characteristics and connection
- Unit 1: Complex Numbers (Solutions) @fsc-part1-kpk:sol
- ===== Unit 1: Complex Numbers (Solutions) ===== This is a first unit of the book Mathematics 11 published... e students will be able to * Recall complex number $z$ represented by an expression of the form $z=a... or of the form $(a,b)$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$. * Recognize $a$ as real p... $z$. * Know condition for equality of complex numbers. * Carry out basic operations on complex num
- Unit 01: Complex Numbers (Solutions) @math-11-kpk:sol
- ===== Unit 01: Complex Numbers (Solutions) ===== This is a first unit of the book Mathematics 11 publishe... e students will be able to * Recall complex number $z$ represented by an expression of the form $z=a... or of the form $(a,b)$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$. * Recognize $a$ as real p... $z$. * Know condition for equality of complex numbers. * Carry out basic operations on complex num
- Unit 02: Matrices and Determinants (Solutions) @math-11-kpk:sol
- the students will be able to * Recall complex number $z$ represented by an expression of the form $z=a... or of the form $(a,b)$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$. * Recognize $a$ as real par... f $z$. * Know condition for equality of complex numbers. * Carry out basic operations on complex numbers. * Define $\bar{z} = a —ib$ as the complex conjugat