Formula Section
</p> <p>f_k = f(x_k),: x_k = x^*+kh,: k=frac{N1}{2},dots,frac{N1}{2}
binom{n}{k} = frac{n!}{k!(nk)!}
where h is some step.
Then we interpolate points ((x_k,f_k)) by polynomial
P_{N1}(x)=sum_{j=0}^{N1}{a_jx^j}
Its coefficients {a_j} are found as a solution of system of linear equations:
left{ P_{N1}(x_k) = f_kright},quad k=frac{N1}{2},dots,frac{N1}{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.
NonTerminating Decimal Fraction:
A decimal number that has no end after the decimal point.
NonTerminating Repeating Decimal Fraction
In nonterminating decimal fraction, some digits are repeated in same order after decimal point.
NonTerminating NonRepeating Decimal Fraction.
In nonterminating 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.
 Nonterminating recurring (repeating) decimals are rational numbers.
 Nonterminating and nonrecurring (repeating) decimals are irrational numbers.
Note:
 Repeating decimals are called recurring decimals.
 Nonrepeating decimals are called nonrecurring 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=55=0
 20+20=0
 1010=0
 \frac{2}{3}+\frac{2}{3}
 \frac{2}{3}+\left ( \frac{2}{3} \right) =0
 \sqrt{2}+\left( \sqrt{2} \right) =0
 9.59.5=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
 a5=b or b=a5
 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
x3=7
Add 3 on Both sides
x3+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 ac=bc
Example
x=5 then x2=52
x+3=7
Subtract 3 from Both sides
x+33=73
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