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Formula Section

 

 </p> <p>f_k = f(x_k),: x_k = x^*+kh,: k=-frac{N-1}{2},dots,frac{N-1}{2}

binom{n}{k} = frac{n!}{k!(n-k)!}

where h is some step.
Then we interpolate points ((x_k,f_k)) by polynomial

   P_{N-1}(x)=sum_{j=0}^{N-1}{a_jx^j} 

Its coefficients {a_j} are found as a solution of system of linear equations:
   left{ P_{N-1}(x_k) = f_kright},quad k=-frac{N-1}{2},dots,frac{N-1}{2}

 

 

This is (e=lim_{ntoinfty} left(1+frac{1}{n}right)^nlim_{ntoinfty}frac{n}{sqrt[n]{n!}} )

frac{1+frac{a}{b}}{1+frac{1}{1+frac{1}{a}}}

Introduction to Real Numbers

Set of Natural Numbers

N=\{1,2,3,4,\dots\}

Set of Whole Numbers

W=\{0,1,2,3,4,\dots\}
Set of Integers Z=\{0, \pm1, \pm2, \pm3, \pm4, \dots\}
OR
W=\{\dots,-4,-3,-2,-1,0,1,2,3,4, \dots\}

Rational Numbers

The word Rational means “Ratio”. A rational number is a number that can be expressed in the form of \frac{p}{q} where p and q are integers and  Rational numbers is denoted by Q

Set of Rational Numbers

Q=\left\{ \frac{p}{q}|p,q \in Z,q \neq0 \right\}

Irrational Numbers

The word Irrational means “Not Ratio”. Irrational number consists of all those numbers which are not rational. Irrational numbers are denoted by Q^/ .

Real numbers

The set of rational and irrational numbers is called Real Numbers. Real numbers is denoted by R Thus QUQ^/=R
Note:
All the numbers on the number line are real numbers.

Terminating Decimal Fraction:

A decimal number that contains a finite number of digits after the decimal point.

Non-Terminating Decimal Fraction:

A decimal number that has no end after the decimal point.

Non-Terminating Repeating Decimal Fraction

In non-terminating decimal fraction, some digits are repeated in same order after decimal point.

Non-Terminating Non-Repeating Decimal Fraction.

In non-terminating decimal fraction, the digits are not repeated in same order after decimal point.
Decimal Representation of Rational and Irrational Numbers.

    • All terminating decimals are rational numbers.
    • Non-terminating recurring (repeating) decimals are rational numbers.
    • Non-terminating and non-recurring (repeating) decimals are irrational numbers.

Note:

  • Repeating decimals are called recurring decimals.
  • Non-repeating decimals are called non-recurring decimals

Properties of Real Number
The set R of real number is the union of two disjoint sets. Thus  QUQ^/=R
Note:
Q \cap Q^/=\emptyset
Real Number System

Closure Property w.r.t Addition
The sum of real number is also a real number. If a, b \in R then a+b \in R
Example:
7+9=16 Where 16 is a real number.
Closure Property w.r.t Multiplication
The Product of real number is also a real number. If a, b \in R then a.b \in R
Example:
7×9=16 Where 63 is a real number.
Commutative Property w.r.t Addition:
If a, b \in R then a+b=b+a
Example:
7+9=9+7
16=16
Commutative Property w.r.t Multiplication
If a, b \in R then a.b=b.a
Example:
7.9=9.7
63=63
Associatve Property w.r.t Addition:
If a, b, c \in R then a+(b+c)=(a+b)+c
Example:
2+(3+5)=(2+3)+5
2+8=5+5
10=10

Associatve Property w.r.t Multiplication
If a, b, c \in R then a(bc)=(ab)c
Example:
2(3×5)=(2×3)5
2(15)=(6)5
30=30

Additive Identity:
Zero (0) is called Additive identity because adding “0” to a number does not change that number. If we add 0 to a real number, the sum will be the real number itself.
If a \in R there exists 0 \in R then a+0=0+a=a
Example:

  • 3+0=0+3=3
  • -5+0=-5
  • 9+0=9
  • \frac{2}{3}+0=\frac{2}{3}
  • 9.5+0=9.5

Multiplicative Identity
1 is called Multiplicative identity because multiplying “1” to a number does not change that number. If we add 1 to a real number, the product will be the real number itself. If a \in R there exists 1 \in R then a.1=1.a=a

Example:

  • 3×1=1×3=3
  • -5×1=-5
  • 9×1=9
  • \frac{2}{3} \times 1=\frac{2}{3}
  • 9.5×1=9.5

Additive Inverse

When the sum of two numbers is zero (0). If we add a real number to its opposite real number, the result will always be zero (0). If a in R there exists an element a^/ then a+a^/=a^/+a=0 then a^/ is called additive inverse of a
OR
a+(-a)=-a+a=0
10+(-10)=-10+10=0

Example:

  • 3+(-3)=0
  • -5+5=5-5=0
  • -20+20=0
  • 10-10=0
  • -\frac{2}{3}+\frac{2}{3}
  • \frac{2}{3}+\left ( -\frac{2}{3} \right) =0
  • \sqrt{2}+\left(- \sqrt{2} \right) =0
  • 9.5-9.5=0

Multiplicative Inverse
When the product of two numbers is 1.
If we multiply 1 to a real number, then the product will be the real number itself. If a in R there exists an element a^{-1} then a.a^{-1}=a^{-1}.a=1 then a^{-1} is called multiplicative inverse of a.
OR
a.\frac{1}{a}= \frac{1}{a}.a=1 10. \frac{1}{10}=\frac{1}{10}.10=1


Examples:

  • 5. \frac{1}{5}=1
  • -3 \times \frac{1}{-3}=1
  • -3 \left ( \frac{1}{-3} \right)=1
  • \frac{1}{3} \times 3 =1
  • \frac{5}{3} \times \frac{3}{5} =1
  • \left (\frac{5}{3} \right) \left (\frac{3}{5} \right) =1
  • \left (-\frac{5}{3} \right) \left (-\frac{3}{5} \right) =1
  • \left (\frac{-5}{3} \right) \left (\frac{-3}{5} \right) =1
  • \sqrt{2} \left ( \frac{1}{\sqrt 2} \right) =1
  • 9.5 \left ( \frac{1}{9.5} \right) =1

Distributive Property of Multiplication over Additon

Propeties of Equality of Real Numbers

Reflexive Property
Every real number or value is equal to itself. e.g. a=a which means that a itself equal to a
Example

  • 5=5
  • \frac{1}{5}= \frac{1}{5}
  • -3 =-3
  • -3.8 =-3.8
  • \sqrt{2} = \sqrt{2}
  • 5.9+\sqrt{2} = 5.9+\sqrt{2}
  • x+y=x+y

Symmetric Property

By interchanging the sides of an equation doesn’t effect the result. e.g. a=b then b=a does not effect the result.
In other words,

  • Left side equal to right side of an equation
  • Right side equal to left side of an equation

Example
9+7=16 then 16=9+7

  • x=16 or 16=x
  • x+y=z or z=x+y
  • x+2=z or z=x+2
  • a-5=b or b=a-5
  • 5.9+\sqrt{2} =x or x = 5.9+\sqrt{2}

Note

If x=y then x may be replaced by y or y may be replaced by x in any equation or expression

Symmetric Property may not worked in some cases such as Subtraction or Division

Trasnsitive Property

If a equal to b under a rule and b equal to c under the same rules then  a equal to  c is known as transitive property. e.g.  a=b and  b=c then  a=c

 

Example

x+y=z and z=a+b then x+y=a+b

x=5+y and 5+y=a+b then x=a+b

Addition Property

If we add the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then a+c=b+c

Example

x=5 then x+2=5+2

x-3=7

Add 3 on Both sides

x-3+3=7+3

x=10

Subtraction Property

If we Subtract the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then a-c=b-c

Example

x=5 then x-2=5-2

x+3=7

Subtract 3 from Both sides

x+3-3=7-3

x=4

Multiplication Property

If we Multiply the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then a \times c=b \times c

Example

x=5 then x \times 2=5 \times 2

\frac{x}{3}=7

Mutiply 3 on Both sides

\frac{x}{3} \times 3=7 \times 3

x-=21

Division Property

If we Divide the same number or expression on both sides of an equation, the equation does not change means both the sides remain equal. e.g. a=b then \frac{a}{c} = \frac{b}{c}

Example

x=5 then \frac{x}{3} = \frac{5}{3}

2x=24

Divide Both sides by 2

\frac{2x}{2}=\frac{24}{2}

x-=12

 


Second law of motion