A2 1 linear fxns

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Relations ,[object Object],[object Object]

Can be represented graphically by a set of ordered pairs (x, y).,[object Object],[object Object]

Linear Functions ,[object Object],[object Object]

A solution to a linear function is a point on the line.,[object Object]

Graphing ,[object Object],[object Object]

For a linear function, all these points together create a line.,[object Object],[object Object]

Graphing ,[object Object],Solve the equation for y. Choose a few x-values to plug in and find the corresponding y-values.

Graphing ,[object Object],Solve the equation for y. Choose a few x-values to plug in and find the corresponding y-values. Plot these points.

Graphing ,[object Object],Solve the equation for y. Choose a few x-values to plug in and find the corresponding y-values. Plot these points. Connect with a straight line.

The intercept is where it crosses the axes.,[object Object]

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Complex analysis and its application 2.Contents,Complex number Different forms of complex number Types of complex number Argand Diagram Addition, subtraction, Multiplication & Division Conjugate of Complex number Complex variable Function of complex variable Continuity Differentiability Analytic Function Harmonic Function Application of complex Function 3.Complex Number,For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and-5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits. All numbers are imaginary (even "zero“ was contentious once). Introducing the square root(s) of minus one is convenient because all n-degree polynomials with real coefficients then haven roots, making algebra "complete"; it saves using matrix representations for objects that square to-1 (such objects representing an important part of the structure of linear equations which appear in quantum mechanics ,heat,diffusion,optics,etc) .The hottest contenders for numbers without purpose are probably the p-adic numbers (an extension of the rationales),and perhaps the expiry dates on army ration packs. 4.Complex Number is defined as an ordered pair of real number X & Y and is denoted by (X,Y) It is also written as 𝒛=𝒙,𝒚=𝒙+𝒊𝒚,where 𝑖^2=−1 𝑥 is called Real Part of z and written as Re(z) Y is called imaginary part of z and written as Im(z). -If R(z) = 0 then 𝑧=𝑖𝑦, is called Purely Imaginary Number. -If I(z) = 0 then 𝑧=𝑥, is called Purely Real Number. -Here 𝑖can be written as (0, 1) = 0 ±1𝑖 Note:-−𝒂= 𝑎−1=𝑖𝑎 -If 𝑧=𝑥+𝑖𝑦is complex number then its conjugate or complex conjugate is defined as 𝒛=𝒙−𝒊𝒚. 5.DIFFERENT FORMS OF COMPLEX NUMBER Cartesian or Rectangular Form :-𝑧=𝑥+𝑖𝑦 Polar Form :-𝑧=𝑟(cos𝜃+𝑖sin𝜃) 𝑜𝑟 𝑧=𝑟∠𝜃 Exponential Form :-𝑧=𝑟𝑒^𝑖𝜃 MODULUS & ARGUMENT OF COMPLEX NUMBER Modulus of complex number (|z|) OR mod(z) OR 𝑟=√(𝑋^2+𝑌^2 ) Argument OR Amplitude of complex number (𝜃) OR arg (𝑧) OR amp(z)=tan^(−1)⁡(𝑥/𝑦) 6.Argand Diagram Mathematician Argand represent a complex number in a diagram known as Argand diagram. A complex number x+iy can be represented by a point P whose co–ordinate are (x,y).The axis of x is called the real axis and the axis of y the imaginary axis. The distance OP is the modulus and the angle, OP makes with the x-axis, is the argument of x+iy. 7.Addition of Complex Numbers Let a+ib and c+id be two numbers, then (a+ib)+(c+id)=(a+c)+i(b+d) Procedure: In addition of complex numbers we add real parts with real parts and imaginary parts with imaginary parts. 8.Subtraction of Complex Numbers Let a+ib and c+id be two numbers, then (a+ib)-(c+id)=(a-c)+i(b-d) Procedure: In subtraction of complex numbers we subtract real parts w

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A2 1 linear fxns

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A2 1 linear fxns