Decimal Number

NUMBER SYSTEM
Definition
It defines how a number can be represented using distinct symbols. A number can be represented differently in different systems, for instance the two number systems (2A) base 16 and (52) base 8 both refer to the same quantity though the representations are different.

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers.
A computer can understand positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
A value of each digit in a number can be determined using
The digit
The position of the digit in the number
The base of the number system (where base is defined as the total number of digits available in the number system).

1. Decimal Number System
The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands and so on.
Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position, and its value can be written as
(1x1000)+ (2x100)+ (3x10)+ (4xl)
(1x103)+ (2x102)+ (3x101)+ (4xl00)
1000 + 200 + 30 + 1
1234
As a learner you should understand the following number systems which are frequently used in computers.
S.N.
Number System & Description
1
Binary Number System
Base 2. Digits used: 0, 1
2
Octal Number System
Base 8. Digits used: 0 to 7
4
Hexa -Decimal Number System
Base 16. Digits used: 0 to 9, Letters used: A- F

2. Binary Number System
Characteristics
Uses two digits, 0 and 1.
Also called base 2 number system
Each position in a binary number represents a 0 power of the base (2). Example 20
Last position in a binary number represents a x power of the base (2). Example 2x where x represents the last position - 1.
Example
Binary Number: 101012
Calculating Decimal Equivalent:
Step
Binary Number
Decimal Number
Step 1
101012
((1 x 24) + (0 x 23) + (1 x 22) + (0 x 21) + (1 x 20))10
Step 2
101012
(16 + 0 + 4 + 0 + 1)10
Step 3
101012
2110
Note that: 101012 is normally written as 10101.
3. Octal Number System
Characteristics
Uses eight digits, 0,1,2,3,4,5,6,7.
Also called base 8 number system
Each position in a octal number represents a 0 power of the base (8). Example 80
Last position in a octal number represents a x power of the base (8). Example 8x where x represents the last position - 1.
Example
Octal Number: 125708
Calculating Decimal Equivalent:

Step
Octal Number
Decimal Number
Step 1
125708
((1 x 84) + (2 x 83) + (5 x 82) + (7 x 81) + (0 x 80))10
Step 2
125708
(4096 + 1024 + 320 + 56 + 0)10
Step 3
125708
549610
Note that 125708 is normally written as 12570.

4. Hexadecimal Number System
Characteristics
a. Uses 10 digits and 6 letters, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F.
b. Letters represents numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15.
c. Also called base 16 number system
d. Each position in a hexadecimal number represents a 0 power of the base (16). Example 160
e. Last position in a hexadecimal number represents a x power of the base (16). Example 16x where x represents the last position - 1.
Example
Hexadecimal Number: 19FDE16
Calculating Decimal Equivalent:
Step
Binary Number
Decimal Number
Step 1
19FDE16
((1 x 164) + (9 x 163) + (F x 162) + (D x 161) + (E x 160))10
Step 2
19FDE16
((1 x 164) + (9 x 163) + (15 x 162) + (13 x 161) + (14 x 160))10
Step 3
19FDE16
(65536+ 36864 + 3840 + 208 + 14)10
Step 4
19FDE16
10646210

They also include:

Natural numbers
The counting numbers (1, 2, 3,4,5 ...), are called natural numbers. They include all the counting numbers i.e. from 1 to infinity.
Whole numbers
They are the natural numbers and zero. Not all whole numbers are natural numbers, but all natural numbers are whole numbers.
Integers
They are positive and negative counting numbers, as well as zero.
Rational numbers
They are numbers that can be expressed as a fraction of an integer and a non-zero integer.

Real numbers
Thet are all numbers that can be expressed as the limit of a sequence of rationalnumbers. Every real number corresponds to a point on the number line.
Irrational numbers
A real number that is not rational is called irrational.
Complex numbers
Includes real numbers and imaginary numbers, such as the square root of negative one.

LANGRARIAN MULTIPLIER
Definition
It is a strategy for finding the local maxima and minima of a function subject to equality constraints.
In mathematical optimization, the method of Lagrange multipliers (named after Joseph Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.
For instance in Figure 1, consider the optimization problem maximize subject to
We need both and to have continuous first partial derivatives. We introduce a new variable () called a Lagrange multiplier and study the Lagrange function (or Lagrangian) defined by

Where the term may be either added or subtracted. If is a maximum of for the original constrained problem, then there exists such that is a stationary point for the Lagrange function (stationary points are those points where the partial derivatives of are zero, i.e. ). However, not all stationary points yield a solution of the original problem. Thus, the method of Lagrange multipliers yields a necessary condition for optimality in constrained problems.Sufficient conditions for a minimum or maximum also exist.
Introduction
One of the most common problems in calculus is that of finding maxima or minima (in general, "extrema") of a function, but it is often difficult to find a closed form for the function being extremized. Such difficulties often arise when one wishes to maximize or minimize a function subject to fixed outside conditions or constraints. The method of Lagrange multipliers is a powerful tool for solving this class of problems without the need to explicitly solve the conditions and use them to eliminate extra variables.
Consider the two-dimensional problem introduced above: maximize subject to
We can visualize contours of f given by

for various values of , and the contour of given by .
Suppose we walk along the contour line with . In general the contour lines of and may be distinct, so following the contour line for one could intersect with or cross the contour lines of . This is equivalent to saying that while moving along the contour line for the value of can vary. Only when the contour line for meets contour lines of tangentially, do we not increase or decrease the value of — that is, when the contour lines touch but do not cross.
The contour lines of f and g touch when the tangent vectors of the contour lines are parallel. Since the gradient of a function is perpendicular to the contour lines, this is the same as saying that the gradients of f and g are parallel. Thus we want points where and
,
where

and

are the respective gradients. The constant is required because although the two gradient vectors are parallel, the magnitudes of the gradient vectors are generally not equal.
To incorporate these conditions into one equation, we introduce an auxiliary function

and solve

This is the method of Lagrange multipliers. Note that implies .
The constrained extrema of are critical points of the Lagrangian , but they are not local extrema of
Applications ol Langragian multiplier
Economics
Constrained optimization plays a central role in economics. For example, the choice problem for a consumer is represented as one of maximizing a utility function subject to a budget constraint. The Lagrange multiplier has an economic interpretation as the shadow price associated with the constraint, in this example the marginal utility of income. Other examples include profit maximization for a firm, along with various macroeconomic applications.
Control theory

In optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the Hamiltonian, in Pontryagin's minimum principle.
REFERENCES

BINOMIAL THEOREM
It is a theorem that specifies the expansion of a binomial of the form (x+y) to the exponent and as the sum of n=1 terms of which the general term consists of a product of x and y with x raised to the exponent (n-k) and y raised to the exponent k and a coefficient consisting of n! divided by (n-k)!k!
The Binomial Theorem is a quick way (okay, it's a less slow way) of expanding (or multiplying out) a binomial expression that has been raised to some (generally inconveniently large) power. For instance, the expression (3x – 2)10 would be very painful to multiply out by hand. Thankfully, somebody figured out a formula for this expansion, and we can plug the binomial 3x – 2 and the power 10 into that formula to get that expanded (multiplied-out) form.
Binomial
A binomial is a polynomial with two terms
5y3-3
example of a binomial
Exponents

An exponent says how many times to use something in a multiplication.
In this example: 82 = 8 × 8 = 64

An exponent of 1 means just to have it appear once, so you get the original value:
Example: 81 = 8
An exponent of 0 means not to use it at all, and we have only 1:
Example: 80 = 1

Exponents of (a+b)
Now on to the binomial.
We will use the simple binomial a+b, but it could be any binomial.
Exponent of 0.
Exponent of 0
When an exponent is 0, you get 1:
(a+b)0 = 1
Exponent of 3
For an exponent of 3 just multiply again:
(a+b)3 = (a+b)(a2 + 2ab + b2) = a3 + 3a2b + 3ab2 + b3

The Pattern
In the last result we got: a3 + 3a2b + 3ab2 + b3
Now, notice the exponents of a. They start at 3 and go down: 3, 2, 1, 0:

Example: When the exponent, n, is 3.
The terms are: k=0: k=1: k=2: k=3: an-kbk
= a3-0b0
= a3 an-kbk
= a3-1b1
= a2b an-kbk
= a3-2b2
= ab2 an-kbk
= a3-3b3
= b3

Coefficients we have: a3 + a2b + ab2 + b3
But we need: a3 + 3a2b + 3ab2 + b3
We are missing the numbers (which are called coefficients).
As a Formula
In binomial have Pascal's Triangle that helps us solve binomial problems You find the coefficient from Pascal's Triangle

References
Binomial theorem in ancient India". By Amulya Kumar
“Computing Cavalieri's Quadrature Formula by a Symmetry of the n-Cube".by Barth Nils
Binomial Coefficients by Graham, Ronald; Knuth, Donald; Patashnik, Oren Concrete Mathematics by Addison Wesley

EGERTON UNIVERSITY.
NAIROBI CITY CAMPUS.
DEPARTMENT OF COMMERCE
COURSE BACHELOR OF COMMERCE.
Mwangi Damaris Wambui.
CP122/66508/12.
Semester 1,year 1.
Assignment: Business mathematics