Introduction to parallel line extender:

Properties of parallel lines are the under the base of Euclid's parallel property. Two lines in a single plane that not even intersect or meet at any point then it is called as parallel lines. In other words Parallel lines are nothing but “A pair of lines in a plane which do not intersect or meet each other” then they are called parallel lines.

i.e. m1 = m2,

Where, m1 = slope of the first line.

m2 = slope of the second line.

Parallel Line Extender Problem:

Example for parallel line extender: Define the equation parallel to 4y + 4x = 8 and the line extender, with point (6.5,4).

Solution:

Given 4y - 4x = 8 and the point (6.5, 4)

To detect the perpendicular line, we have to find slope first.

For finding the slope, we need to change the given equation into slope intercept form.

4y + 4x = 8

Add 4x on both side,

4y + 4x = 8

- 4x = -4x

4y = -4x + 8

Divide by 4 on both sides,

y = (-x + 2)

The obtained equation is in the form, y = mx + b

So, the slope from the obtained equation m = -1

From generally we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = -1

So, m2 = -1

The line equation is,

(y - y1) = m(x - x1)

(y - (4)) = -1 (x - ( 6.5))

(y - (4)) = (-x + 6.5)

Y - 4 = - x + 6.5

Subtract 4 on both sides,

y = -x + 2.5

Answer: Thus, the parallel line extender is given through the line y = -x + 2.5

Example for Parallel Line Extender:

Example for parallel line extender: Define the equation parallel to 2y - 2x = 4 and the line extender is gets through the point (-2, 0.5).

Solution:

Given 2y - 2x = 4 and the point (4, 3)

To detect the perpendicular line, we have to find slope first.

For finding the slope, we need to change the given equation into slope intercept form.

2y - 2x = 4

Add 2x on both side,

2y - 2x = 4

+ 2x = +2x

2y = 2x + 4

Divide by 2 on both sides,

y = (x + 2)

The obtained equation is in the form, y = mx + b

So, the slope from the obtained equation m = 1

From generally we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = 1

So, m2 = 1

The line equation is,

(y - y1) = m(x - x1)

(y - (3)) = 1 (x - (4))

(y - (3)) = (x - 4)

Y - 3 = x - 4

Subtract 3 on both sides,

y = x - 1

Answer: Thus, the parallel line extender is used to find the line y = x - 1

Introduction to positive divisors

In math divisor is defined as “the number that you are going to divide by”. Divisor of a given integer can also be called as the divisor of a given integer.

Dividend /Divisor = Quotient.

Positive number/ Negative number = negative number

Negative number/Positive number = negative number

Positive/Positive  = Positive

Let us have an example 16/2 = 8 here 2 is a divisor.

16/4= 4 here 4 is a divisor.

16/8= 2 here 4 is a divisor.

In the above shown example the divisors are positive and also known as positive divisors.

The positive divisors when the dividend is 16 are 8, 4, and 2.

Example Problems for Positive Divisors:

Example problem 1:

Find the positive divisors of 28

Solution:

Positive Divisors of 28 are 1, 2, 4, 7, 14, and 28

Where 28 /1 = 28.

28 /2 = 14.

28 /4 = 7.

28 /7 = 4.

28 /14 = 2.

28 / 28 = 1

Hence for 28 we have 6  Positive divisors

Moreover 28 is a composite number

Example problem 2:

Find the positive divisors of 52

Solution

Positive   Divisors of 52 are 1, 2, 4, 13, 26, and 52.

Where 52/1=52.

52/2=26.

52/4=13.

52/13=4.

52/26=2.

52/52=1.

Hence for 52 we have 6   Positive divisors

Example problem 3:

Find the positive divisors of 30

Solution

Positive Divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.

Where 30/1=30

30/2= 15.

30/3=10.

30/5=4.

30/6=5.

30/10=3.

30/15=2.

30/30=1

Hence for 30 we have 8 Positive divisors

Example problem 4:

Find the positive divisors of 100

Solution

The positive divisors of  100 are  1,2,4,5,10,20,25,50,100

Where 100/1=100

100/2= 50.

100/4=25.

100/5=20.

100/10=10.

100/20=5.

100/25=4.

100/50=2.

100/100=1.

Hence for 100 we have 9 Positive divisors

Example problem 5:

Find the positive divisor of 31

Solution

Given that 31 is a prime number

All the prime numbers have only two divisors

The two divisors are 1 and the numbers itself

So the positive divisor of the number 31 is 1, 31 alone.

Finally the divisors of the prime numbers are 1 and the number itself.

Practice Problems for Positive Divisors:

Problem 1:

Find the positive divisor of 63

Solution:

The divisors of 63 is 1, 3,7,9,21,63

So the number 63 has 6 divisors.

Problem 2:

Find the positive divisor of 37

Solution:

The divisors of 37 are 1, 37

Problem 3:

Find the positive divisor of 15

Solution:

The divisors of 15 is 1, 3, 5, 15

Problem 4:

Find the positive divisor of 35

Solution:

The divisors of 35 is 1, 5, 7, 35

Understanding how to graph parabolas

A parabola (math) in co-ordinate geometry is defined as the locus of a point whose distance from a fixed point in the plane and perpendicular distance from a fixed line in the plane (not passing through the fixed point) are equal. The fixed point is called the focus of the parabolic and the fixed line is called the directrix of the parabolic.

These have many applications. Parabola help us to understand the trajectory of a projectile. The reflectors in car head lights or speakers in a sound system or mirrors in a telescope are all parabolic in shape. Parabolic mirrors are also used to harness solar energy. These are only a few examples where parabolas are useful. Parabolic forms find application in various other sciences field as well. Therefore it is of prime importance that we know how to graph a parabola.

Graph the parabola:

Parabola graphs are of two types: (1) horizontal and vertical. The horizontal form is again sub categorized into two types: (a) the ones that open left and (b) the ones that open right. Similarly the vertical forms are also of two types: (a) ones that open up and (b) the ones that open down.

The general equation of a horizontal form is: x = a(y-k)^2 + h, where (h,k) is the vertex of the parabolic. Whether the parabolic form opens left or right would depend on the sign of the term ‘a’. If a is positive, then that opens right and if a is negative the same opens left.

The general equation of a vertical parabolic form is y = a(x-h)^2 + k, where again (h,k) is the vertex of the same. If the parabolic form opens up then the ‘a’ in the above equation would be positive. If ‘a’ in the above equation is negative then the parabolic form would open down.

As we can see from the equation of the para-bola stated above, these equations are quadratic. The standard form of a quadratic equation in x is f(x) = ax^2 + bx + c. This would be a vertical parabolic form. If we were to find the vertex from this equation it would be: h = -b/2a and k = f(h). There to if ‘a’ is negative, these open down and if ‘a’ is positive, the same opens up.

Adding and Subtracting Matrices

A matrix is a rectangular array of symbols, numbers or expressions which were arranged in a sequence of rows and columns. These having only one row (1 x n) is known as a row vector and if it is having only one column (m x 1) is called as a column vector. The forms of these which are of the same size can be subjected to both addition and subtraction by element to element.

Matrix Addition and Subtraction
If in case when two matrices are needed to be either added or subtracted, then there lays a condition. It is the condition that both of these should be of the same order. This means that both of the forms involved in the operation should have the same number columns and rows. The expression generally used to represent the addition and subtraction is as follows.

Zij = Aij  +  Bij . This means that the elements of the ‘i’ row and column ‘j’ of the matrice A and the elements of the ‘i’ row and ‘j’ column of the B are added or subtracted. When we do this, then it will result in a new matrix of the same order in which addition or subtraction was done. This is nothing but Z form which has ‘i’ rows and ‘j’ columns.

Matrix Addition Rules
As we saw earlier about the order of matrices that is to be added or subtracted should have the same order there lays one more rule or necessity for doing addition. When ‘m’ is the total number of rows and ‘n is the total number of columns; then ‘i’ will be greater than or equal to 1 and less than or equal to ‘m’. Also ‘j’ will be greater than or equal to 1 and less than or equal to ‘n’.

Let us see an example for both Matrix addition and Matrix Subtraction.

• Let us consider of A and B matrices having two rows and three columns. The matrice A has elements 1, 2 and 3 in the first row and 4, 5 and 6 in the second row. While the B has elements 3, 4 and 5 in the first row and 7, 8 and 9 in the second row. Now for Addition, we do first row as 1+3, 2+4 and 3+5. Also for second row, we do as 4+7, 5+8 and 6+9. Hence we will get a final part of first row as 4, 6 and 8 and second row with elements as 11, 13 and 15.
• Similarly subtracting instead of adding from the above step, we will get a matrice with first row elements as -2, -2, -2 and second row as -3,-3,-3. This is better known as its subtraction form.

Real numbers

Real Number Definition– As the name says “Real”, real numbers are actually numbers that really exists. Any number that we think of is considered as a real number, be it positive or negative, fraction or decimal. Real numbers are numbers those includes both rational and irrational number. A real number has to have a value. If there is no value to any number then we can call that number as an imaginary number. All integers like -75, 89, and 84 etc. are considered as real numbers. All fractions like 3/5, 7/2, -9/7 are considered as real numbers too.

Decimals along with repeating decimals are also considered as real numbers. A real number can be any positive or negative number. We can plot All Real Numbers on the number line too.  Therefore we can order the real numbers and that we cannot do in case of imaginary numbers. The name of imaginary numbers itself says that they are imaginary so we can just imagine them; they do not have a specified value.
The real numbers can be plotted the same way we plot the integers that is smaller numbers on left and larger numbers on right. So greater the number, more it will be towards right side of number line.So we can call real numbers as all those numbers which are present on number line are termed as real numbers.
Some Real Numbers Examples are pi, 34/7, 5.676767, -1034, 45.87 etc. Some examples of imaginary numbers are square root of -34 or square root of -2, as there is a negative under the root, so the value of this number cannot be found.

Similarly value of infinity cannot be determined too .Hence these numbers are not real numbers and are considered as an imaginary numbers. So real numbers are all integers, fractions decimals and repeating decimals numbers.

We can add, subtract, multiply and divide real numbers just like another numbers. We can perform the operations on real numbers same way as we do them on other numbers.

Now the question arises that Is 0 a Real Number– Zero is considered as an integer and all integers are real numbers. Therefore zero is considered to be as a real number. All Real Numbers Symbol is R which is used by many mathematicians. The symbol R is used to represent the set of real numbers. All real numbers can be seen on the number line but we cannot find imaginary numbers on that.

Learning to divide Mixed Numerals

We know that a number in the form of P/Q is called a fraction. Both P and Q are integers. The fractions can be proper fractions, improper fractions and sometimes even mixed fractions. Mixed fractions are also known as mixed numerals or mixed numbers. The operations of addition, subtraction, multiplication and division can be performed on these fractions. Now we will learn the process of dividing mixed fractions to get the final answer. This can be done in a few simple steps. The first step would be to convert the mixed fraction into an improper fraction. Improper fraction is nothing but a fraction in which the denominator must be less than numerator. After this step again the question arises how to divide mixed fractions to arrive at the final answer. The second step is to get the divisor. Divisor is the number or the fraction by which the mixed fraction has to be divided. After getting the divisor, the numerator of divisor is made its denominator and vice versa and multiplied with the improper fraction we got from the mixed fraction given.

The product got after the multiplication is the answer. Now the question how do I divide fractions would have been cleared in the minds of many. This is a very simple process as we understand it. We get division fractions in the process. After finding the reciprocal of the divisor and multiplying with the improper fraction we get a product. This has to be simplified to the lowest terms possible. Only then we get the right answer. Simplifying to lowest terms can be done by dividing the numerator and the denominator by a common number so that both the numerator and denominator are fully divisible by that number. This process has to be continued till the numerator and denominator in the lowest forms and they are not further divisible. This process is done to simplify the fraction and not keep it as large numbers. This will help in the calculations as we need deal with large numbers. Also the time consumed would be less. Lesser the time consumed more will be efficiency. The process of division takes bit more time than other operations of addition, subtraction and multiplication. This is because division involves multiplication as well. This can be overcome by doing more and more practice. Once we have enough practice we can bring down the time required to solve a problem.

Introduction to Boolean logic calculator

Boolean logic algebra is the algebra of two values. These values are usually taken to be 0 and 1, as we shall do here, although F and T, false and true, etc. Boolean logic algebra are also a common uses of Boolean logic calculator. More generally Boolean algebra is algebra of values from any Boolean algebra as a model of the laws of Boolean algebra. In Boolean logic calculator first enter the expression then then enter the values of x and y. now press the calculate button then the answer will be produced.

Basic Operations Involved in Boolean Logic Calculator:

in Boolean logic calculator the operations mentioned below are involved automatically ,    

After values, the next ingredient of any algebraic system has its operations. Whereas elementary algebra was based on numeric operations multiplication x • y, addition x + y, and negation −x, Boolean logic algebra calculator is customarily based on logical counterparts to those operations, namely conjunction x • y (AND), disjunction x + y (OR), and complement or negation ¬x (NOT). In electronics, the multiplication is represented by a AND, an addition is represented by OR, and the NOT is represented with an over bar.

Conjunction was the closest of these three to its numerical counterpart, in fact on 0 and 1 it is multiplication. As the logical operation conjunction of two propositions is true when both propositions are true, and otherwise is false.

Disjunction works almost similar to addition in the boolean logic calculator, with one exception: the disjunction of 1 and 1 is neither 2 nor 0 but 1. Thus the disjunctions of two propositions are false when both propositions are false, and otherwise are true.

Logical negation does not work like numerical negation in the calculator. Instead it corresponds to incrementation: ¬x = x+1 mod 2. It shares the common numerical negation the property that applying it twice returns the original value: ¬¬x = x, just as −(−x) = x.

Some of the identities and operations involved in boolean logic is also in boolean logic calculator as,

(1a) x • y = y • x                                     (1b) x + y = y + x                   (1c) 1 + x = 1

(2a) x • (y • z) = (x • y) • z                      (2b) x + (y + z) = (x + y) + z

(3a) x • (y + z) = (x • y) + (x • z)             (3b) x + (y • z) = (x + y) • (x + z)

(4a) x • x = x                                          (4b) x + x = x

(5a) x • (x + y) = x                                  (5b) x + (x • y) = x

(6a) x • x1 = 0                                        (6b) x + x1 = 1

(7) (x1)1 = x

(8a) (x • y)1 = x1 + y1                            (8b) (x + y)1 = x1 • y1

Problem Based on Boolean Logic Calculator:

Problem 1: Find the value for x.y = y.x

x y x.y y.x
0 0 0 0
0 1 0 0
1 0 0 0
1 1 1 1