Triangle Inequality Lec 4, Complex Analysis For BS & MS Classes by Maqsood Ali Abbas

Very important Triangle inequality
#ComplexAnalysis
#BS
#MS

[saim]

Saima parveen (1008)
Q:1 lower bound:
The lower bound is the smallest value that would round up to the estimated value
Q:2 Upper bound:
The upper bound is the smallest value that would round up to next estimated value
Q:3 Triangle inequality
In mathematics the triangle inequality states that
For any triangle the sum of lenght of any two sides must be greater than or equal to the length of remaining side

Q:4 What is modulus
The length of the line segment is called the modulus of the complex number and is denoted as |z|
Q:5 conjugate:
The conjugate of a complex number is the number with an equal real part and an imaginary part eqaul in magnitude but opposite in sign

[mubasharrehmani]

Mubashar Hussain (1011)...
QNO1...
Given a number z =X+iy complex conjugate simply by changing the sign of the imaginary part of the complex
number....z=x-iy...
QNO2...
Definition of Modulus of a Complex Number: Let z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x2+ y 2) is called the modulus or absolute value of z...
QNO3
Triangle inequality may b define as
the sum of the lengths of any two sides must be greater than or equal to the length of the other side.
QNo4
The subset of set A which is greater than or equal to every element of A is called upper bound of set ....
QNo5...
The subset of set A which is less than or equal to every element of A is called lower bound of set ...

[anayasubhan]

Insha Akhtar 1009
Q#1 The length of line segment is called modulus of complex number and is denoted by |z|.
Q#2 The complex conjugate of complex number is the number with equal real part and imaginary parts equal in magnitude but oppsite in sign.

[motivationalstar]

Amina liaqat (1006)
Definition of Modulus of a Complex Number: Let z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x2+ y 2) is called the modulus or absolute value of z (or x + iy). Sometimes, |z| is called absolute value of z.

[motivationalstar]

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude, but opposite in sign.
For example: z= a+ bi then conjugate of complex no.is a- bi

[abdulrehmanjb]

Ayesha Riaz 1042 (Morning )
Q 1. Congujate of complex no.
The complex conjugate of complex number is the number with an equal real and imaginary parts but opposite in sign.
Q.2. Modulus of complex number.
Let Z= a+bi be a complex number. Then the Modulus of a complex number Z denoted by |Z| is defined to be the non negative real number.
Q.4. Upper bound of any set.
If every number in the set is less than or equal to the bound the bound is an upper bound.
Q.5. Lower bound of any set.
If every number in the set is greater than or equal to the bound the bound is a lower bound.

[taimourakhtar]

Taimour Akhtar (1012) (Morning)
Q#1 In any two complex numbers, if only the sign of the imaginary part differ then, they are known as complex conjugate of each other.
Conjugate of a complex number
z = a + ib, denoted by z¯, is defined as Conjugate of z1 = 3 + 5i is z1¯ = 3 - 5i

Q#2
Modulus of a Complex Number:
Let z = x + iy where x and y are real and i = √-1. Then the non negative square root of (x2+ y 2) is called the modulus or absolute value of z (or x + iy).
Q#3 Answer:
Triangle enquilites in words The sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c
Q#4 what is upper bound or lower bound Lower bound?
The lower bound is the smallest value that would round up to the estimated value.
Upper bound:
The upper bound is the smallest value that would round up to the next estimated value.
Example:
in {3, 5, 11, 20, 22} 3 is a lower bound, and 22 is an upper bound.

[rimshazahid]

Rimsha Zahid 2204
Q1. Conjugate of Complex Number:
In Mathematics, the complex conjugate of complex number is the number with an equal real part and imaginary part equal in magnitude, but opposite in sign. Given a complex number z=a+bi (where a and b a real numbers), the complex conjugate of z, often denoted as z(bar), is equal to a-bi.
Q2. Modulus Of Complex Number:
Let z=x+iy where x and y are real and i=√-1 .Then the non negative square root of (x∧2+y∧2) is called the modulus or absolute value of z (or x+iy).
Modulus of complex number z =x+iy, denoted by mod(z) or |z| or |x+iy|.
Q3. The Triangle Inequality:
The sum of any two sides of a triangle is greater than or equal to the third side ; in symbols, " a+b≥c".
Q4. Lower Bound:
A value that is less than or equal to every element of a set of data.
Q5. Upper Bound:
A value that is greater than or equal to every element of a set of data.

[masoomshah]

Masoom Ali (1044)
Q/1)The magnitude of line sagment is called modulus.
Q;2)The complex numbur in which the sign of imaginary part is inverse that of the original complex numbur, is called conjugate complex numbur.
Q;3)Let z1 and z2  be vectors. Then the triangle inequality is given by
Geometrically, the right-hand part of the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.

[Muhammadirfan-vtcd]

Muhammad Irfan
2212
Q1: conjugate
Ans: conjugate is formed by changing the sign between two terms in a binomial.
Q2: modulus
Ans: The length of the line segment is called the modulus of the complex number and is denoted |z|.

[emanfatimaemanfatima]

Eman Fatima 2253 (Evening)
Q:1 Definition Conjugate of complex number
Given a complex number Z=x+iy, if we put-i in place of I we get x-iy which is usually written as z bar which is called conjugate of complex number.
Actually z bar is the mirror image of z.
Q:2Mudulus of complex number.
The modulus of complex number is the distance from the origin of the point representing the nmbr.
Q:4 Triangle inequality in words.
The absolute value of sum of two complex number never eceed the sum of their modulus.
Q:5 Lower bound and upper bound of complex number
Lower bound; a value that is less than or equal to every element of a set of data
Upper bound; a value that is greater than or equal to every element of a element of a set of data.

[qandyqueen]

QANDEEL TARIQ(1004)Morning
Q1:conjugate of a complex no: conjugate of a complex no is the no with an equal real part and an imaginary part equal in magnitude but opposite in sign.Given a complex no z=a+bi(where a and b are real nos)the conjugate of a complex no is often denoted as Z– mod i.e.z–=a-bi
Q2:let z =x+yi where x and y are real and i=Saqure root -1.then the non negative square root of (X^2+y^2)is called modulus of complex no.It is also called absolute of z.
It is denotes by|z|
Q3:Triangle inequality states that for any triangle the sum of length of any two sides must be greater than or equal to the length of remaining side.
Q4:lower bond:A value that is less than or equal to every element of a set of data.
Example{3, 6, 9, 19, 25}here 3 is the lower bond.
Q5:upper bond:A value that is greater than or eqaul to every element of a set if data.
Example:{3, 6, 9, 19, 25}25 is the upper bond .
Note:A quick way to calculate upper and lower bands is to halve the degree of accuracy specified, then add this to the rounded value for the upper bound and subtract it from the rounded value for the lower bond.

[asadasif]

Asad Asif 1029
Ans1:
Conjugate of complex no. Is equal to the real and opposite of imaginary part of comlex no. (It is the image of comlex no.)
Ans2:
Modules of complex no.is the magnitude of complex no.
Ans3:
Uper bound:
The no.of st which is greater than oe equal to every no. Of other elements of set
Lower bound:
The no. Which is smaller than or equal to every no.of set
Ans4:
The sum of lengths of two sides of triangle is equal or greater than remaing side

[arifarafique]

Arifa 1005 Morning
Q1: Conjugate:
In algebra conjugate is where you change the sign (+ to -, or -to+) in the middle of the two terms
Q:2 Modulus:
The absolute value of a real or complex number |a|
Q:3 Triangular inequality:
In mathematics the triangular inequality states that for sum of the length of any two sides must be greater than or equal to the length of the measuring side.
Q:4 Lower bound:
A value that is less than or equal to every element of a set of data
Q:5 Upper bound:
A value that is greater than or equal to every element of the set of data.
Example: in {3, 5, 11, 20, 22} 3 is a lower bound and 22 is a upper bund but be careful! 2 and lower numbers are also a lower bound because it is less than 3 and 23 and above are upper bound because it is greater than 22

[nomanrasheed]

Noman Rasheed 2207 (evening)
Ans 1) conjugate of complex number:
the complex conjugate has the same real part as z and the same imaginary part but with the opposite sign.
To find the complex conjugate of 45+3i we change the sign of the imaginary part. Thus the complex conjugate of 45+3i is 45 - 3i.
Ans 2) Modulus of complex number:
The length of the line segment is called the modulus of the complex number and is denoted |z|. The angle measured from the positive real axis to the line segment is called the argument of the complex number, denoted arg(z) and often labelled θ. The modulus and argument can be calculated using trigonometry.
Ans 3) Triangle inequality:
the right-hand part of the triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.
Ans 4) Lower bound of a set:
A value that is less than or equal to every element of a set of data. Example: in {3, 5, 11, 20, 22} 3 is a lower bound.
Ans 5) upper bound of set:
A value that is greater than or equal to every element of a set of data.

Example: in {2, 7, 11, 20, 25} 25 is the upper bound.

[jawadrana]

M.jawad 1019
Q.1
A complex number Z = x+iy, if we put -i in place of i we get x-iy which is usually written as Z bar which is called conjugate of Z actually Z is the mirror image of Z bar.
Q.2
|Z|=r=x^2+y^2 is called the modulus of z.
Q.3
The whole modulus of two complex number is always less than and equal to the sum of their respective modulus.
Q.4
An upper bound of a subset S of some pre ordered set is an element of K which is greater than or equal to every element of set S
Q.5
An lower of S is defined to be an element of K which is less than or equal to every element of S.

[natashashafaqat]

Natashashafaqat1026
1) by changing the sign of imaginary part we can find the conjugate
2)the length of line segment is called modulus of complex number
3)triangle in equality state that sum of lengths of two sides of triangle is equal or greater than the length of remaining side
4)the lower bound is the smallest value that would round up the estimated value
5) upper bound is the smallest value that would round up the next estimated value

[NimraHafeez-pwzz]

Nimra Hafeez 2203
Q#1:
Ans: A conjugate is formed by changing the sign btween two terms in bionomial x+y or x-y.
Q#2:
Ans: Modulus of complex nmbr is the length of directed line segment drawn from origion of complx plane to plan (a, b).
Q#3:
Ans: The triangle inequality states that for any triangle the sum of length of two sides must be greater than or equal to length for remaining side.
Q#4:
Ans: An upper bound or majorant of a subset S of some preordered set is greater or equal to every element of S.
Q#5:
Ans: A lower bound or minorant of S is defined to be an element of K which is less than or equal to every element of S.

[Muhammadirfan-vtcd]

Muhammad Irfan
2212
Q3: triangle inequality.
Ans: for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
Q4: lower bound of set.
Ans: Any number that is less than or equal to all of the elements of a given set.

[muhammadanas]

Muhammad Anas (1043) Morning
Q#01: Conjugate of a complex number:-
The conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude, but opposite in sign.
Q#02: Modulus of a complex number:-
A modulus of a complex number is the length of the directed line segment drawn from the origin of the complex plane to the other point in the plane.
Q#03: Triangle inequalities:-
The length of the side of the triangle corresponding to the vector Z1+Z2 cannot be longer than the sum of the length of the remaining two sides.
Q#04: Upper bound of a set:-
A value that is greater than or equal to every element of a set of data is called upper bound of a set.
Q#05: Lower bound of a set:-
A value that is less than or equal to every element of a set of data e is is called lower bound of a set.
Example: {3, 5, 11, 20, 22}, here 3 is the lower bound and 22 is the upper bound of this set.