Adding and Subtracting Positive and Negative Numbers

Positive and Negative numbers are important concepts in arithmetic. A number that is greater than 0 (zero) is called positive number. On the other hand, any number that is less than 0 are called negative numbers. A positive number can be written with the plus sign in front of the number or just as a number. For example: 1, 2, 3, 4 …….n. A negative number is written with the minus sign in front of the number. For example: -1, -2, -3, -4…..-n. There are different rules of operations for positive and negative numbers. The basic sets of rules are the addition and subtraction rules of arithmetic for positive and negative numbers. Let’s have a look at the same in this post.

Adding Positive and Negative Numbers
Positive + Positive = Positive: The sum of two positive numbers is equal to a positive number. For example: (2 girls’ vest ) + (3 girls’ vest) = 5 girls’ vest.
• Positive + Negative = Negative: The sum of one positive and one negative number results a negative number. For example: (2 toddler shoes) + (- 5 toddler shoes) = - 3 toddler shoes.
• Negative + Negative = Positive: The sum of two negative numbers results a positive number. For example: (- 2 online baby stores ) + (- 2 online baby stores) = 4 online baby stores.
• Negative + Positive = Negative: The sum of one negative and one positive number results a negative number. For example: (-3 apples) + (5 apples) = - 2 apples
Subtracting Positive and Negative Numbers
• Negative – Positive = Negative: When a negative number is subtracted by a positive number, the result is negative. For example: (-8baby shoes) – (3 baby shoes) = -11 baby shoes
• Positive – Negative = Positive: When a positive number is subtracted by a negative number, the result is positive. For example: (8 baby slings) – (- 5 baby slings) = 13 baby slings.
• Negative – Negative = Negative: When two negative numbers are subtracted, the result is negative. For example: (- 5 oranges) – (- 5 oranges) = - 10 oranges.
These are the rules of addition and subtraction for positive and negative numbers.

Difference of Sets

Difference of Sets is the operation on set(s). if there is a st A and a st B then st difference A – B  gives the elements of st A that are not in st B. similarly by Difference Set B – A gives the elements of st B that are not in st A.

As shown in above Venn diagram A – B is the closed curve with blue outline. This shows clearly that A– B is composed of those elements which do not belong to st B. set diffrnce A – B can also be stated as st A – (A∩B).  A∩B is the part which is common in A and B. so A – B is said to be that part of A which has exclusion of common part of A and B or which excludes common part with st B.

A – B = {x: x ∈A,and  B}
For example if there exists a st S1 {1, 2, 3, 4, 8, 10} and st S2 {2, 6, 7, 4, 8, 5} then S1 – S2 gives the elements of S1 that are not in S2 {1, 3, 10}. As you see this difference does not contain any element of S2 and also it contains only those elements of S1 that are not in st S2 or that are uniquely in st S1. Similarly S2 – S1 or above example gives a st = {6, 7, 5}.

Symmetric Difference of Sets is another operation done on sts as explained below:
Given two sts A and B then by symmtric Difference Sets A and B gives those elements of A and B which are either in A st or in B st but not in both st. We can also say that it gives the union of two sts but excluding the common element of them or intersection of them. This operation is represented as: A ∆ B = (A ∪B)-(A ∩ B ) as shown in diagram given below:

   
Example: if st A = {2, 3, 4, 5, 7} and st B = {3, 6, 7, 8, 9} then A ∆ B = {2, 4, 5, 8, 9}. Elements 3, 6, 7 are not there as they are common in both the sts. Another way to do it is find union and intersection of both sts and subtract intersection from union as:  A ∪B  = {2, 3, 4, 5, 6, 7, 8, 9} and A ∩ B = {3, 6, 7}. So (A∪B)-(A∩B ) ={2,4,5,8,9}
R Set Difference of B and A gives elements of B but not of A.

Tenses – Past, Present and Future

Tense is one of the most important concepts in English Grammar learning. Tense in English grammar is a category that refers to a situation in time. This concept of time is classified into three types of tenses, namely: Present Tense, Past Tense and Future Tense. Let’s have a look at each of the type of tenses.
Present Tense:
Present tense is a type of tense that refers to the current time. In simple terms, the tense that is a verb referring to the action or expression of present is called present tense. For example:
• She is listing down return gift ideas for the party. (Here, it is present tense because ‘she’ is listing down the return gift ideas in the present time.)
• Sana is exploring the online infant shopping India collection. (Here, it is present tense because ‘Sana’ is exploring the online infant shopping India collection in the present time.
• I am eating. (Here, ‘I’ am eating at the present time.)
Past Tense:
Past tense is a type of tense that is a verb referring to an action or expression in the past. For example:
• Mary’s cousin bought nappy pads online sometimes back. (Here, the sentence is referring that Mary’s cousin has bought nappy pads online in the past.)
• I had a heavy lunch this afternoon. (Here, it is past tense as the sentence is referring to the lunch happened in the past)
• I was eating. (Here, I was eating at the past time.)
Future Tense:
Future tense is a type of tense that expresses actions or state something in future. For example:
• India will be having many new online stores in the near future. (Here, the verb is referring to a situation in future.)
• My niece will start going to school from next year. (Here, niece will go to school in future.)
• I will eat. (Here, I will eat in future)
These are the three types of tenses in English Grammar.

Introduction to rectangle images:

Rectangle is one of the 2D object and also quadrilateral regular polygon. It contain four sides. Opposite of the sides are equal in rectangle Image. Rectangle image having four sides. Each side are intersected with 90 degree Two sides are lengths of the rectangle. Two sides are width of the rectangle. In rectangle images Opposite sides of lengths are equal and also opposite sides of width are equal. Rectangular changes should be based on length of the rectangle and width of the rectangle.

Basic Concepts of Rectangle Images:

 From the above image AB,BC,CD,DA These are Sides of the rectangle Image.

AB and CD are Parallel sides of the rectangle Images

AB and CD These are Length of the Rectangles

BC and AD These are Width of the rectangle Images

AC and BD Are Diagonals of the Rectangle Images (Rectangle Image Contains Two Diagonal AC and AD .Two Diagonals Are Equal length o f Rectangle Image)

 Area and Permeter of Rectangle:

Area of the rectangle:Area of the rectangle is Prodouct of Length and  width

Area of rectangle= Length*Width

Perimeter of the rectangle=2(Length + Width)

Example Problems in Rectangle Images:

Ex 1: Find the area of a rectangle whose length is 10m and width is 25 m .

Solution: Area of the rectangle is Product of Length and  width

Area of rectangle= Length*Width    =10*25,  Area = 2500 m^2  

Ex 2: Find the Perimeter of the rectangle image?

Solution: Perimeter of the rectangle=2(Length + Width)

From the given image Length =20,width= 5

Perimeter = 2(5+20)    =2(25)    =50cm

Ex 3: Find the area of a rectangle whose length is 1/2m and width is 8/15 m .

Solution: The area of a rectangle is multiplication of Their length and Width

Length= `1/2` in

Width=`8/15`

Area = Length * width = `1/2 xx 8/15 = 8/30 = 4/15`

Area= `4/15` in^2

Ex 4: Find the Perimeter of the rectangle image,Their length is 12 and width is 2

Solution: Perimeter of the rectangle=2(Length + Width)

From the given image Length =12,width= 2

Perimeter = 2(12+2)      =2(14)  =28cm

What to understand by diameter of a circle

What is a circle?
In a third grader’s language, a circle is a closed figure with curved side or no sides. A circle is defined by its centre and its radius. The centre of a circle is the point exactly in the middle of the circle such that every point on the circle is at the same distance from the centre of the circle. The radius of the circle is the distance between the centre of the circle and any point on the circle. Since all points on a circle are at the same distance from the centre of the circle, there can be infinitely many radii (plural of radius) of a circle.

What is the diameter of a circle?
When we draw two radii from the centre of a circle to any two different points on a circle, an angle is formed such that the two radii are the rays of the angle. This angle can be an acute angle, an obtuse angle, a right angle, a straight angle or also a reflex angle. When this angle is a straight angle, that is, when the angle at the centre subtended by the two radii is 180 degrees, the two radii form a straight line that passes through the centre of the circle and touches two points on the circle. Such a line is called the diameter of the circle. In other words, a diameter of a circle is a line that passes through the centre of the circle and touches any two points on the circle as well.

Diameter of a circle can also be defined in terms of the chord of a circle. A chord of a circle is a line segment that joins any two points on the circle. If a chord is such that it passes through the centre of the circle then it is called the diameter of the circle.
Just as in radii, a circle can have infinitely many diameters as well. By the definition of a diameter stated above, we also see that all the diameters would pass through the centre of the circle. Therefore we can say that all diameters of a circle are concentric.

How to find the diameter of a circle?
From the definition of a diameter above we saw that the length of the diameter would be two times that of the radius. Therefore the diameter of a circle formula can be written like this:
d = 2r
Where, d  is the diameter of the circle and r is the radius of the circle.

Simplifying Algebraic Expressions

Algebraic expressions are the representations in order to neglect writing in terms of words of relation between various terms. There are two variations in algebraic expressions which can be said as the rational and irrational algebraic expressions. A rational expression is nothing but an algebraic expression which can be written in a quotient of polynomials, for example x2 + 2x + 4. Meanwhile, an irrational expression is the one that in which is not rational, for example square root of (x+4).

Algebra Simplifying Expressions
In general, the algebraic expressions contain numbers and alphabetic symbols. Simplifications of some lengthy algebraic expressions are needed in order to shorten and solve the problem. While making that simplification, an equivalent expression would be finally arrived which is simpler than the previous. It obviously means that the simplified expression is smaller than the normal.

How to simplify Algebraic Expressions?

• There are no standard procedures for the simplification of algebraic expressions since there can be so many methods can be made which differ from person to person while doing. But, those can be grouped in some form as three types as follows.
• Expressions which can be simplified without any kind of preparation immediately.
• Expressions which requires preparation before doing the simplification.
• Expressions which cannot be simplified to any form.
• Also there are some rules are methods that can be used while doing the simplification. Before going into the example for simplifying, we shall see some of the rules while doing them.
• While doing the addition of fractions:  When the denominator is common for all the fractions, then add all the numerators and sum it and divide it by the common denominator.
• Simplify the expression as much as possible to the extent.
• Order of Expression: When there are confusingly operations were given, for example 8+4(2+3)2-7, then the order of doing simplification as follows. 1) Simplify the terms inside the parenthesis 2) Evaluate the powers and exponents if it is there, 3) Multiply or divide 4) Add or subtract. 

Examples of simplifying Algebraic Equations

• The below some examples would give an idea of how to simplify expressions. 2 x+3y+6-3x-2y-2+4yx. This expression can be easily simplified by adding all the like terms and combining finally. Hence it can be rewritten as (2-3+4) x + (3-2) y + (6-2) and finally it gives 3x+y+4.
• To simplify the expression of previously said example of 8+4(2+3)2-7, after first process we will get, 8+4(5)2-7, then 8+4(25)-7, then 8+100-7 and finally we will get answer as 101.

Properties of numbers

While counting any quantity we use numbers 1,2,3….and so on. In various calculations using numbers the four basic operations used are the addition, subtraction, multiplication and division. Based on these operations there are properties of numbers which make the calculations simpler. These properties lay the foundation to work with different equations and hence it is important to get familiar with them. To the question What are the Properties of Numbers, we can say that the basic property number are the commutative property, associative property, distributive property and Identity.

Commutative Property: A given operation is said to be commutative if when the numbers are interchanged the value of the result remains the same. When numbers are added in whatever order the result remains unchanged and hence addition operation is commutative.

For Example:  3+4=7 also 4+3=7.
When numbers are multiplied in any order the result remains unchanged and hence the multiplication operation is commutative.

For Example: 3x7=21 also 7x3=21. Subtraction operation and division operation are not commutative as the result changes when numbers are interchanged; 7-4=3 but 4-7 = -3; 12/3=4 but 3/12=1/4, here the results are different when numbers are interchanged.

Associative Property: A given operation is said to be associative if when the change in the grouping does not change the result.
When numbers are added the grouping of numbers does not change the result and hence addition operation is associative. For example: 3+(4+5) is same as (3+4)+5. When numbers are multiplied the grouping of numbers does not change the result and hence multiplication operation is associative. For example: 2x(4x5) is same as (2x4)x5.
Subtraction operation and Division operation are not associative.

Distributive Property: This property helps to multiply the number outside with each of the terms inside the parenthesis thus helps to remove the parenthesis. For example: 3(a+b)= 3.a + 3.b= 3a+3b; (x-2)(y+3) each of the terms in the first parenthesis is multiplied with each of the terms in the second parenthesis giving the required product, xy+3x-2y-6.

Identity Property: When a zero is added to any number the result is the same number and hence zero is called the additive identity. For example: 3+0=3. When a number is multiplied by one the result is always the same number and hence one is multiplicative identity. For example: 4x1=4

The other property of numbers is multiplicative inverse, the product of any number and its inverse is always one, a x 1/a=1; 1/a is the multiplicative inverse of a. Zero property, when any number is multiplied with zero the product is zero, ax0=0