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

Addition and Its Properties

Addition is one of the four basic mathematical operations that refer to combination of collections of objects towards forming a larger number. For example: Mother bought two Pigeon baby products for Rs. 220 and three baby toys from Fisher Price India brand for Rs. 350. How much she paid in total for Max online shopping. Adding, Rs. 220 + Rs. 350 = Rs.  570. Thus, mother paid Rs. 570 in total for Max online shopping for the baby. This is called addition operation in mathematics. There are five properties of addition, namely

• Identity Property
• Commutative Property
• Associative Property
• Property of Opposites
• Property of Opposite of a Sum

Identity Property: According to the identity property of addition, there is a unique real number zero for every real number a, b or c etc. Zero is referred as the identity elements in addition. For example: Mother bought one powder (suppose a) from pigeon baby products. Mathematically implementing identity property of addition, a + 0 = a and 0 + a = a.

Commutative Property: According to the commutative property of addition, for all real numbers, the order of addition does not change the result. For example: Mother bought 3 baby toys (suppose a) from Fisher Price India brand and 7 baby products (suppose b) from Pigeon baby brand. Mathematically, 3 + 7 = 10 and 7+3 = 10. Therefore, a + b = b + a

Associative Property: This property of addition states that when three real numbers are added, the grouping or association doesn’t change the result. Example is similar to the previous one. Therefore, (a + b) + c = a + (b + c).

Property of Opposites: This property of addition states that against every real number, suppose a, there is a unique real number, i.e, -a. A number and its opposite are called additive inverses and the addition of additive inverses is equal to zero. Therefore, a + (-a) = 0.

Property of Opposite of a Sum: This property of addition states that the opposite of a sum of real numbers is equal to the sum of the opposites. For example: - (2+3) = (-2) + (-3) = 5. That is, - (a + b) = (-a) + (-b).
These are the properties of addition.

Double Bar Graphs

Double Bar Graphs, also called as Double Bar Chart, is the type of bar graph that helps in finding the relationship between two set of data. These are used to compare the data between same value and within the group of value from time to time. Double Bar Graphs also has all the advantage and attributes as it like the regular bar graph. Double Bar Graphs may be created in horizontal or vertical position as per the given data.

Double Bar Graphs Properties

• We have to follow the instruction to construct the double bar graphs. The instructions are given below.
• The frequency scale must start from zero and go up in equal steps in x-axis and y-axis.
• Bars should have equal width.
• Two types of data should have different colors and the abbreviation for the color was given separately in the graph sheet.
• Both of the axes were labeled and they should name as per the given data.

Advantage and Difference between Single Bar Graph and Double Bar Graphs

• Double Bar Graphs are easy to construct with the help of given set of data.
• It is easy to understand and easy to find the difference between two set of data.
• Data group in the double bar graph is represented by two bars as comparing with single bar graph

Illustration for Double Bar Graphs

Let us see in detail about the double bar graph with an example. Construct a double bar graph for the given set of data.


               Mathematics   Science
Student 01      85          70  
Student 02      95          75  
Student 03      65          80  
Student 04      82          70  
Student 05      88          85  
Student 06      90          80  

• Double bar graph was drawn for six students to compare their marks in mathematics and science.
• To construct the graph, first we have to decide about the title of the graph.
• Then choose the vertical or horizontal bars.
• We name the x-axis as students and y-axis as marks.
• Choose the scale in the y-axis as per the given marks.
• Put label on the graph and finally draw the bars.

Sine graph generator

Sine graph is the periodic with period 2`pi` .They wriggle back and forth between -1 and 1 in a smooth way. Sine starts at 0 and goes up to 1 .Sine graph generator produce the sinusoidal wave. The sine function graph usually how to find phase sift, period, amplitude for the equation

Sine graph generator following conditions produced y = sinx ,  Domain (− ∞, ∞) , Range [− 1, 1] , Principal domain (−π / 2,π / 2)

Sample Problem for Sine Graph Generator:

Example 1:

 Draw the amplitude for given question y=a sinx for various values of  a.

Solution:

For sine is how far the graph gets away from the x-axis, so it is called the amplitude.

Y=a sinx ,  where a=1,2,3 0.5,0.2

See the given  Sine graph  generator

For sine is how far the graph gets away from the x-axis, so it is called the amplitude.

Y=a sinx    a=2 , -2

See the given  Sine graph  generator

y= 2 sinx, -2sinx

Example 2:

 Draw the amplitude for given question y= sinx for various values of  a.

Solution

y=sinx where x values follows
 x =1, -1
see the sine graph generator

Example 3:

Show that Periods other than 2pi is a little harder; because there is a lot that the graph does in the course of its period. P=2`pi` /b

Y=sinbx where b=1,2,3,0.5

solution:

See the given  Sine graph  generator

Practice Problem for Graphs of Sine Generator:

1. Graphing the amplitude for given question y=d sin2x for various values of d

Answer: d values 1,2,3,1.5,4

2. Graphing the period of given question y= sinbx for various values of b

Answer: b values 1,2,3,1.5

3. Graphing sine periods for y= sinbx values of b and d are;

Answer: b=1,2,3,0.5

4. Graphing the amplitude for given question y=t sin3x for various values of t

Answer: t values 1,2,1.5 0.5,0.3.