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.

Binary System and Introduction

What are Binary Numbers
In the word Binary, ‘Bi’ means two, so a system of numbers which involves only two digits is called binary system. The two binary numbers used are ‘1’ and ‘0’. The binary system works in the same way as the decimal system which consists of ten digits which are 0 to 9, in case of binary system there are only two possible digits 1 and 0. Any number which is expressed in the binary system would be a combination of the two digits, 1 and 0. It is mainly useful in the field of technology.  In an electrical circuit we have only two options which are ‘on’ and ‘off’ and hence based on this the binary system plays an important role in all the communications which take place inside a computer. In a decimal system the place values are ones, tens, hundreds, thousands, ten thousands and so on. These values can be written as power of 10 which would be 10^0, 10^1, 10^2, 10^3, 10^4 and so on. In the same way the binary system starts from the right and proceeds towards left the difference being the base is 2.

Binary Numbers Chart can be given as below
2 raised to power zero = 2^0=1
2 raised to power one = 2^1=2
2 raised to power two= 2^2=4
2 raised to power three=2^3=8
2 raised to power four = 2^4=16
2 raised to power five= 2^5=32
2 raised to power six = 2^6=64 and so on, in general it can be considered as 2 raised to power x = 2^x
For example, let us consider a binary number 1011 and convert it into decimal form
Binary conversion: 8   4   2    1
      Decimal form:  1   0   1    1
1011(base 2) = 1x(8) + 0x(4) + 1x(2) + 1x(1) = 8+0+2+1=11(base 10)

List of Binary Numbers are as follows
0000=0; 0001=1; 0010=2; 0011=3; 0100=4; 0101=5; 0110=6; 0111=7; 1000=8; 1001=9; 1010=10 and so on.
Subtracting Binary Numbers
While subtracting binary numbers the following are the facts to be remembered:
0 – 0 =0 ;  1 – 0 = 1; 1 – 1= 0 and 10 – 1 is equivalent to 2 – 1 = 1 in binary system
Subtract 1011 from 10101.

0110 0110 1
-  1   0 1 1
__________
          1  0 1 0        
(1 -1 =0; 10 – 1 =1;0 – 0 = 0; 10 – 1 = 1 and 0 = 0 which gives 1 0 1 0)
Binary number      10101   -    1011 = 1010
Decimal equivalent   21       -       11   =10

Absolute Values

In algebra and in vectors we come across numbers in negative directions although the magnitudes of such quantities have no signs or supposed to be positive.  Let us take a closer study.



What is Absolute Value in Math
Let us consider the numbers 5 and -5. Algebraically, they are not same. But when you plot these numbers on a number line, they are no doubt on the opposite sides of the origin 0. But they are at the same distance from the origin. That is -5 is also 5 units away from the origin on a number line. Hence considering only the fact of ‘how far’, we can say that the absolute value 0f -5 is 5.
Thus we can define that an absolute value of a number is only its physical value and hence it is always referred as positive. Symbolically the absolute value of a variable ‘x’ is denoted as lxl and it is always equal to + x.

Finding Absolute Value
As explained earlier an absolute value of a number cannot be negative. Thus, the easiest method of finding absolute value of a number is just consider the number only, ignoring the sign before it. That is for finding absolute value of -10, just consider only the number 10. That is l-10l = 10.

Limit of Absolute Value
Since the absolute value of a variable is always positive, the upper limit of the absolute value is infinity. However, because an absolute value can never be negative, it cannot cross below 0. Therefore, the lower limit of absolute value is 0. Thus the limit of absolute value of a variable can be expressed as [0, ∞).

Absolute Value Practice
The concept of absolute value is important. For example, A is situated 10 miles from his office and B is situated 5 miles from the same office but exactly in the opposite direction. Now to calculate the distance between A and B, we cannot algebraically say 10 + (-5) = 5 miles or 5 + (-10) = -5 miles. Here you need to apply the absolute value practice and say the distance is l10l + l5l = 10 + 5 = 15 miles.

Properties of Absolute Values
The important properties of absolute values are,
lxl = x for x ≥ 0, but = -x for x < 0.
If lf(x)l = a then there can be two cases. That is f(x) = a and f(x) = -a.
Suppose lf(x)l + a = b and if a > b, then there is no real solution to the equation.

How to Draw Line Plot?

What is a Line Plot in Math?
The definition of a line plot states that it is a graphical representation of frequency of occurrence of each and every data on a data line. Line plot is graphically represented using some marks such as x or dot.

How to do a Line Plot?
A line is drawn with the data values marked in it as we mark values in the x axis of the graph. The number of occurrences of the data values is marked above the number line individually as the variable x to make a Line Plot.  So the number of “x” marked vertically above each data value shows the number of occurrences of the data value in the given data set. Thus this gives the details of how frequently a data value occurs in the data set.

Example of a Line Plot
Let us consider the marks scored by Students of grade VII in their exam.  Let us assume that there are 10 students in the class and the maximum mark they can score is 100. The marks scored by the ten students are 90, 99, 76, 89, 55, 86, 98, 68, 40, and 35 respectively.

Note that the school follows the grade system as shown below:
• The marks 90-100 will lie in grade A
• The marks 80-89 will lie in grade B
• The marks 70-79 will lie in grade C
• The marks 60-69 will lie in grade D
• The marks 50-59 will lie in grade E
• The marks below 50 are considered as ‘fail’ and they will lie in grade F.

Now let us map the students’ marks in the above grade range:
• The marks lying in grade A are 90, 99, and 98
• The marks in grade B are 89, and 86
• The mark in grade C is 76
• The mark in grade D is 68
• The mark in grade E is 55
• The marks in grade F are 40, 35

From this analysis, we can see that three students have scored grade A, two of them have scored grade B, one student each have scored the grade C,D and E and two students have failed and obtained grade F.  Now, let us mark the line plot with the values A, B, C, D, E, and F:

x
x x x
x x x x x x

<--------------------------------------------------------->
| | | | | |
A B C D E F
Grade
The line plot drawn above gives the graphical representation of the frequency of the occurrence of data in every grade.

Examples of Complementary Angles

What are Complementary Angles?
Two angles are said to be complementary angles, if the sum of the two angles measure 90 degrees.  Thus we can say that two complementary angles together from a right angle measuring 90 degrees. But these two angles need not necessarily be adjacent angles i.e. next to each other.

Example of Complementary Angles
One of the best examples of complementary angles can be seen in right angled triangle.  In a right angled triangle, one angle is the right angle. It is a fact that the sum of all the three angles in a right angle is equal to 180 degrees. Thus, it is clear that the sum of the other two non-right angles will be equal to 90 degrees.  This means that the two non-right angles in a right angled triangle are complementary angles. In other words, we can say that these two angles complement each other.

Some of the other examples of complementary angles are:
55 degrees, 35 degrees
40 degrees, 50 degrees
67 degrees, 23 degrees

How to Solve Complementary Angles?
Now let us see how to find complementary angles.  As we know the definition, finding complementary angles is very easy by applying the definition. If the value of one angle is given, then another angle complementary to that angle is found out by subtracting the given value from 90 degrees.

Example 1:
Consider two angles which are complementary to each other.  If one of the angles is 48 degrees, find the other angle.

Solution: If x is the unknown angle, then 48 added to x will give 90 degrees.  Therefore, x is given by subtracting 48 from 90. i.e., X = 90-48 = 42 degrees.
Therefore, the other angle is 42 degrees.

Some Complementary Angles Problems will be in the form of slightly confusing word problems.

Example 2:
If one of the complementary angles is six more than twice the other angle, find the angles.

Solution:  If the variable Y is considered as one angle measure, then, as per the given statement, the other angle is given by 2Y+6. We know that the sum of the two complementary angles is 90 degrees.  So,
Y+2y+6 = 90
3y = 90-6
Therefore the value of Y will be obtained if 84 are divided by 3. Thus, the value of Y will be 28. If y = 28, then the second angle will be (2*28) + 6 which results in the value 62. Thus, the measures of the angle are 28 and 62.