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Number System

Decimal Representation of Real Numbers

Terminating Decimals

A Decimal having finite number of digits in its decimal part. These are Rational numbers.

Example

• \(434.8888881\)
• \(686.67482\)
• \(749.6428\)

Non Terminating Decimals

A decimal having infinite number of digits in its decimal part. These are non rational numbers.

• \(3.145291....\)
• \(46.2394737423....\)

Recurring Decimals or Periodic Decimals

A decimal in which one or more digits repeat indefinitely. These are Rational Numbers.

Example

• \(3.1311313131\)
• \(1.1211211211\)

Real Numbers

Properties

Property	Addition	Multiplication
Closure Law	\(\forall x,y\in R, x+y\in R\)	\(\forall x,y \in R, x\cdotp y \in R\)
Associative Law	\(\forall x,y,z\in R, x+(y+z)=(x+y)+z\)	\(\forall x,y,z\in R, x(yz)=(xy)z\)
Identity Law	\(\forall x \in R, \exists0\in R\) such that \(x+0=0+x=x\), 0 is identity element of addition.	\(\forall x \in R, \exists1\in R\) such that \(x.1=1.x=x\), 1 is identity element of addition.
Inverse Property	\(\forall x\in R, \exists(-x)\in R\) sucth that \(x+(-x)=(-x)+x=0\)	\(\forall x\in R, \exists(x^{-1})\in R\) sucth that \(x(x^{-1})=(x^{-1})x=1\)
Commutative Law	\(\forall x,y\in R, x+y=y+x\)	\(\forall x,y \in R, xy=yx\)

Equality

Property	Definition
Reflexive Property	\(\forall x\in R, x=x\)
Symmetric Property	\(\forall x,y\in R, x=y\implies y=x\)
Transitive Property	\(\forall x,y\in R\) \(x=y\land y=z\implies x=z\)
Additive Property	\(\forall x,y,z\in R, x=y\implies x+z=y+z\)
Multiplication Property	\(\forall x,y,z\in R x=y\implies x.z=y.z\)
Cancellation Property wrt addition	\(\forall x,y,x\in R, x+z=y+z\implies x=y\)
Cancellation Property wrt Multiplication	\(\forall x,y,x\in R, x.z=y.z\implies x=y\), \(z\rlap{/}{=}0\)

Inequality

Property	Definition
Trichotomy Property	\(\forall x,y\in R\) either \(x=y\) \(x<y\) \(x>y\)
Transition Property	\(\forall x,y,z\in R\) \(x>y \land y>z\implies x>z\) \(x<y \land y<z\implies x<z\)
Addition Property	\(\forall x,y,z\in R\) \(x>y\implies x+z>y+z\) \(x<y\implies x+z<y+z\)
Multiplication Property	\(\forall x,y,z\in R\) \(x>y\implies x.z>y.z\) (\(z>0\)) \(x>y\implies x.z<y.z\) (\(z<0\))

Fraction

Property	Definition
Principle for equality of fractions	$$ \frac{a}{b}=\frac{c}{d} \iff ad=bc $$
Rule for Product of fractions	$$ \frac{a}{b} \cdotp \frac{c}{d}=\frac{ac}{bd} $$
Golden rule of fractions	$$ \frac{a}{b}=\frac{ka}{kb}\;\;(k\rlap{/}{=}0) $$
Rule for quotient of fractions	$$ \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{ad}{bc} $$

Complex Numbers

• The numbers of the form \(x+iy\) where \(x,y\in R\) are called complex numbers. The complex numbers are denoted by \(C\).
• \(x+iy=(x,y)\), where \(x\) is called real part and \(y\) is called imaginary part of complex number.
• Every real number is complex number with zero as its imaginary part.
• Set of Real number is a special subset of complex number
• \(\sqrt{-a}\times\sqrt{-b}=\) \((i\sqrt{a})\times(i\sqrt{b})=\) \(i^2\sqrt{ab}=-\sqrt{ab}\).
• Each imaginary number (\(ib\)) is complex number(\(a+ib\)) but each complex number(\(a+ib\)) not an imaginary number(\(ib\)).
• Imaginary numbers are also called pure complex numbers.
• Sum of four consecutive power of iota is always zero.
• Sum and product of two conjugate numbers are always real number.

Powers of \(i\):

• \(i=\sqrt{-1}\)
• \(i^2=-1\)
• \(i^3=-i\)
• \(i^4=1\)
• \(i^{4n}=1\) where \(n\in Z\)
• \(i^n+i^{n+1}+i^{n+2}+i^{n+3}=0\), \(n\in Z\)
• \(\frac{1}{i}=i^{-1}=-i\)
• \(i^i=e^{-\pi/2}\)
• \(e^{ix}=\cos x+i\sin x\)

Solving \(i\) Powers

• If power is between \(1-4\) use above rules.
• If power is greater then \(4\) divide it with 4 and use above rules.
• If power is too large like \(10000000004\) add all the digits and follow above rules.

Tip

• If power of \(i\) is even it must be \(1\) or \(-1\).
• If power of \(i\) is odd it must be equal to \(i\) or \(-i\)
• If power of \(i\) is multiple of 4 then it must equal to \(1\).

Operations

Operation	Definition
Equality of Complex Numbers	\(a+ib=c+id\implies a=c\land b=d\)
Addition of Complex Numbers	\(a+ib+c+id=(a+c)+(b+d)i\) \((a,b)+(c,d)=(a+c, b+d)\)
Scalar multiplication of complex numbers	\(k(a+ib)=ka+ikb\)
Multiplication of Complex numbers	\((a+ib)(c+ib)=(ac-bd)+(ad+bc)i\)
Division of Complex numbers	$$ \frac{(a,b)}{(c,d)}=\left(\frac{ac+bd}{c^2 + d^2}, \frac{bc-ad}{c^2 + d^2}\right) $$
Reciprocal or Multiplicative Inverse	$$ \frac{1}{z}=\frac{a-bi}{a^2 + b^2} $$
Square Root	for \(z=a+bi\) $$ \pm\left(\sqrt{\frac{|z|+a}{2}}+ i \sqrt{\frac{|z|-a}{2}}\right)\;\; (b>0) $$ $$ \pm\left(\sqrt{\frac{|z|+a}{2}}- i \sqrt{\frac{|z|-a}{2}}\right)\;\; (b<0) $$
Logarithm	\(\log(z)=\log\vert z\vert+i\;arg(z)\)

Tip

Multiplicative inverse of a Complex number can also be found by: $$ z^{{-1} = \left(\frac{Re(z)}{|z|}2}, -\frac{Im(z)}{|z|^2} \right) $$

Properties

Operation	Addition	Multiplication
Identity Property	\((0,0)=9+0i\)	\((1,0)=1+0i\)
Inverse Property	\(\forall (a,b)\in C\) has additive inverse \((-a,-b)\)	\((a,b)\) as non-zero complex number has multiplicative inverse $$ \left( \frac{a}{a^2 +b^2}, \frac{-b}{a^2 +b^2} \right) $$
Distributive Property	Multiplication is distributive over addition in \(C\) $$ (a,b)[(c,d)\pm(e,f)]=(a,b)(c,d)\pm(a,b)(e,f) $$

Conjugate of Complex Number

• Let \(Z\) denotes a complex number then \(Z=x+iy=(x,y)\) then conjugate of \(Z\) is denoted by \(\bar{Z}\) and is defined as \(\bar{Z}=x-iy=(x,-y)\).

Properties

\(\forall z, z_1, z_2\in C\):

• \(\overline{(\overline{z})}=z\)
• \(z.\overline{z}=|z|^2\)
• \(\overline{z_1+z_2}=\overline{z_1}+\overline{z_2}\)
• \(\overline{z_1-z_2}=\overline{z_1}-\overline{z_2}\)
• \(\overline{z_1.z_2}=\overline{z_1}.\overline{z_2}\)
• $$ \overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}\;\;(z_2\rlap{/}{=}0) $$
• $$ \frac{1}{z}=\frac{\bar{z}}{|z|^2} \;\;(z\rlap{/}{=}0) $$

Modulus & Argument of a Complex Number

Let \(Z=x+iy=(x,y)\) then:

• Modulus of \(Z=|Z|=\sqrt{x^2+y^2}\)
• Argument of $$ Z=arg\;Z=\tan^{-1}\left(\frac{y}{x}\right) $$
• Argument is also called Amplitude.

Properties of Argument

• Nature of argument is just like logarithmic function.
• \(arg(z_1.z_2)=arg(z_1)+arg(z_2)\)
• $$ arg\left(\frac{z_1}{z_2}\right)=arg(z_1)-arg(z_2) $$
• $$ arg(\overline{z})=-arg(z)=arg\left(\frac{1}{z}\right) $$
• \(arg(z.\bar{z})\)
• $$ arg\left(\frac{z}{\bar{z}}\right) =2arg(z) $$
• \(Aarg(z^n)=n\; arg(z)\)

Properties of Modulus

• \(|z|=|-z|=|\overline{z}|=|-\overline{z}|\)
• \(z.\bar{z}=|z|^2\)
• \(|z^n|=|z|^2\)
• \(|z_1.z_2|=|z_1|.|z_2|\)
• \(|z_1/z_2|=|z_1|/|z_2|\)
• \(|z_1|-|z_2|\le|z_1+z_2|\le|z_1|+|z_2|\) (Triangular inequality).

Graphical Representation of Complex Numbers:

Since a complex number is considered as an order pair of real numbers, so we can represent such numbers by points in a xy-plane (Cartesian plan) which is called complex plane or Argand Diagram.

Info

In order pair \((a,b)\)

• \(a\) is called x-coordinate or abscissa
• \(b\) is called y-coordinate or ordinate

Principle Argument of a Complex Number

Let \(\theta\) be the principle argument of a complex number \(x+iy\) defined in range \(-\pi<\theta\le\pi\) such that:

• In 1^st Quadrant $$ \theta=\tan^{-1} \left(\frac{y}{x}\right) $$
• In 2^nd Quadrant $$ \theta=\pi-\tan^{-1} \left(\frac{y}{|x|}\right) $$
• In 3^rd Quadrant $$ \theta=\tan^{-1} \left(\frac{|y|}{|x|}\right)-\pi $$
• In 4^th Quadrant $$ \theta=-\tan^{-1} \left(\frac{|y|}{x}\right)-\pi $$

Polar Form of Complex Numbers

Since every complex number \(x+iy\) can be represented in the form of a point \((x,y)\), so we can express every complex number in the form of polar coordinates \((r,\theta)\), where \(r=\sqrt{x^2+y^2}\) and \(\theta=\tan^{-1}(y/x)\)

• \(x+iy=r\cos\theta+ir\sin\theta\)
• To convert polar coordinates into Cartesian coordinates replace \((r, \theta)\) into \((r\cos\theta, r\sin\theta)\)

De Moivre's Theorem

\[ (\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta,\;\;\forall n\in Z \]

• It can be used to solve powers of Complex Numbers.

Applications

• $$ (Z)^{{1/n} = r}{1/n} \left(\cos\frac{\theta+2\pi k}{n} + i\sin\frac{\theta+2\pi k}{n} \right) $$ where \(k=0,1,2,...(n-1)\). This formula is used to find the nth root of a complex number.
• \(Z_1.Z_2=r_1r_2\left(\cos(\theta_1+\theta_2)+i\sin(\theta_1+\theta_2)\right)\)
• $$ \frac{Z_1}{Z_2}=\frac{r_1}{r_2}(\cos(\theta_1-\theta_2)+i\sin(\theta_1-\theta_2)) $$
• Complex numbers do not hold order axioms.

Locus of a Complex Number

A Complex number can form following in a Cartesian Plane:

• Circle if \(|z|=1\)
• Parabola if \(k=|z\pm z_1|\)
• Circle if $$ \frac{z-z_1}{z-z_2}\rlap{/}{=}1 $$
• Line if $$ \frac{z-z_1}{z-z_2}=1 $$
• \(|z-z_1|+|z-z_2|=k\)
  • Ellipse if \(k>|z_1-z_2|\)
  • No locus if \(k<|z_1-z_2|\)
  • Line if \(k=|z_1-z_2|\)
• \(|z-z_1|-|z-z_2|=k\)
  • No locus if \(k>|z_1-z_2|\)
  • Hyperbola if \(k<|z_1-z_2|\)
  • Line \(k=|z_1-z_2|\)

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