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

Numbers in Hexadecimal form

It was the time of representation of things using something else, in short a metaphor which is used to represent things. Numbers has so many forms and Hexadecimal Number is one in that line of representation of numbers. Hexadecimal Number System is usually the numbers from 0 – 9 and A, B, C, D, E, F which represents 10, 11, 12, 13, 14, 15 respectively. The hexadecimal numbers are the numbers with a base 16. The base or the subscript to avoid confusion, similarly we will be using the subscript 10 to denote the decimal number. An example for hexadecimal number is 125AB16. The numbers or values corresponding to each hexadecimal number in terms of decimal, binary or octal in a chart like representation are the Hexadecimal Number Chart. This chart will help us in easily identifying the value of the number in decimal form which every single can understand, it can be named as a compiled chart.

Now let us try to answer the question of How to convert Hexadecimal to Decimal. To do the same we have to follow the steps below which will give the solution: First of all the place value of decimal numbers should be mastered, unit place value is considered as 0, tens place value is considered as 1 and so on. Then hexadecimal should be written with equal space. Then number starting from the left should be multiplied by 16 raised to the power of the place value it is assigned. Finally all the numbers should be added according to their place value.

Let us consider a problem which will give a better grab, Convert the hexadecimal number B432 into decimal number. B = 11*16^3 = +45056, 4 = 4*16^2 = +1024, 3 = 3*16^1 = +48, 2 = 2*16^0 = + 2. The answer is  4613010

Thus the above problem has helped us in understanding the way to convert hexadecimal to decimal. The Hexadecimal Numbering System is not easy for a layman to read or understand the way it, the conversion is to enable every single. The next concept will be Adding Hexadecimal Numbers, to make this concept clear let us consider the following example: ABC + 12A = BE6

The above addition cannot be done when the number which results to more than 16, in that case the number should be subtracted from 16 and a one should be carried over to the next adding digits to be added and goes on.

Vision Statement Examples

Introduction to stem and leaf plot

We are having lot of methods to show the data’s.  We can use the graphs, charts and tables for showing the data. Stem and leaf plot is one of the method which is used for showing large amount of data. It is like the histogram. The main difference between stem and leaf plot and histogram is we can understand the mean, mode and median from this. We will see some examples for back to back stem and leaf plot.

Terms for back to back Stem and Leaf Plot:

If we are having large number of data we will use the back to back stem and leaf plot. Just like the series of data’s. Normally a series mean it will contain more number of data’s. We can analyze the data’s using the back to back stem and leaf plot. In this we have to identify the data’s using its place value. Here the largest place value is known as stem and the other values are leaf. We already know about end to end connection method. Back to back stem and leaf method is like that only. In this we will write the leaves in both side of the stem. In the left side of the stem we are having the rounded leaves and right side we are having the truncated leaves.

Sample Problem for back to back Stem and Leaf Plot:

Back to Back Stem and Leaf Plot problem 1:

First we will see how to make a stem plot for a value consider the value 89

Solution:

Stem     is 8 and the leaf is 9.

Likewise we have to construct the back to back stem and leaf

Here we will see how to make a back to back stem and leaf plot

Let us consider a value 149.

So for this value the rounded leaf is 15 and the truncated leaf 14

Here the stem is 1

So the Back to Back Stem and leaf plot is

Rounded leaf (left)                       Stem                    Truncated leaf (right)

15                                               1                                         14

Back to Back Stem and Leaf Plot problem 2:

Draw the back to back stem and leaf plot for the following data's: 159, 260, 398, 401, 931

Solution:

Let us take the above set in ascending order. 159, 260, 398, 401, 931.

The back to back stem and leaf plot will be like the following


Rounded leaf stem Truncated leaf
16 1 15
26 2 26
40 3 39
40 4 40
93 9
93

Parts of Speech

Parts of Speech are a very important lesson in English grammar. It is a linguistic category of words that is classified based on the behavior of the lexical term. There are eight parts of speech in English grammar – Noun, Pronoun, Verb, Adjective, Adverb, Preposition, Conjunction and Interjection.

Noun: Noun is a part of speech that refers to a concrete entity. Nouns are names of people, places, and things and so on. For example: Johnson & Johnson is the most popular brand for kid’s clothes. Here ‘Johnson & Johnson’ is the name of the brand and a noun in the sentence.

Pronoun: Pronoun is a part of speech that is used as a substitute of noun. As for example: Today is Tina’s birthday and her father gave her a Barbie doll. In this sentence ‘her’ is a pronoun that is used instead of the noun Tina.

Verb: Verb is a part of speech that signifies an activity or a process being performed or done by the noun. As for example: Tina is playing with her Barbie doll. Here, the activity Tina is performing is ‘playing’ and therefore playing is the verb.

Adjective: An adjective is a part of speech that qualifies or modifies a noun or pronoun by stating something about it. An adjective generally precedes the noun or pronoun that it modifies. As for example: Tina is a beautiful girl. Here ‘beautiful’ is an adjective as it is describing the noun Tina.

Adverb: An adverb is a part of speech that modifies an adjective, verb or other adverb. As for example: Mother quickly bought two tops for girls from the market. Here quickly is modifying the verb bought and therefore quickly is the adverb.

Preposition: A preposition is a part of speech that describes a relation between other word and phrases in a sentence. As for example: Mother bought tops for girls from the nearest kid’s store. Here, from is the preposition.

Conjunction: A conjunction is a part of speech that joins two words or phrases. As for example: Ram and Hari are good friends. Here, and is the conjunction.

Interjection: An interjection is an exclamation used to denote emotions. As for example: Alas, the old man is dead. Here, Alas is the interjection that denotes the emotion of sadness.

Countably infinite

Countably infinite on the face of it appears to be a contradiction, which it is not.  In everyday language we may be using the term countable to signify a countable (and hence finite) number. But in mathematical terms countably infinite has a very specific and well defined meaning. In order to understand what countably infinite means,  we first need to understand certain concepts and a little about the history of counting.

Countably Infinite :history of Counting

Primitive man had no use for counting large numbers. While hunting or fighting other tribes he found it enough to indicate one, two or more. This sense of one, two or many is also found among the animals. You will almost never find a lonely animal picking a fight with a large group, while it will willingly fight a one to one battle with another animal. If you observe carefully, you will find that at times a lonely animal does fight two after an initial hesitation. This tendency shows that animals too do have a primitive sense of counting and numbers.

With the development of civilization various systems for counting developed, which have ultimately evolved as numbers the way we know them. The concept of infinity also came into existence with this evolution and the meaning of " infinite " was taken as something that cannot be counted.

Finite and Infinite

Concepts of finite and infinite became known and initially none imagined that there could be different degrees of  " infinite"  too. Researches on number theory, set theory and analysis have proved otherwise. Thus we know today that there are finite and infinite numbers but among the infinite numbers there too are various degrees .  Thus, in present day mathematics infinite cannot be just treated as a number that cannot be counted.

Definition of Countably Infinite

Countably infinite in mathematics means a set of elements which can be mapped one to one on to the set of natural numbers. In other words for which a one to one correspondence can be found between all its elements and natural numbers without skipping any natural number and without assigning either two elements to the same natural number, or assigning two natural numbers to the same element..

Not all infinite sets are countably infinite. It requires a very high degree of skills in mathematics and logic to understand how. But it has been established logically and mathematically that not only a higher degree of infinite exists beyond countably infinite, but for any degree of infinite, there exists one which is beyond it and hence larger.

Dividing Rational Numbers

To review, rational number is any number that can be written as a fraction of integers. We shall understand by using example, 3 and 4 are the integers so 3 divided by 4 would be considered as a rational number. Fractions as we know are rational numbers and so are whole numbers. Let us take 3 divided by 1 is 3 and is the same thing so dividing rational numbers using the multiplicative inverse.

Dividing Rational Numbers or How to Divide Rational Numbers – Rational numbers are divided using the multiplicative inverse. When we divide fractions, we find the multiplicative inverse of the divisor; often it is also known as reciprocal. Reciprocal is when we replace the numerator by denominator and denominator by numerator then the resultant fraction. For example: - For a fraction 7/8, its reciprocal will be 8/7.

To Divide Rational Numbers, we find the multiplicative inverse or reciprocal of divisor. For example, if we have to divide 2/5 by ¾ then ¾ is considered as dividend and 2/5 as the divisor. So in this case reciprocal of divisor will be 5/2. Now instead of dividing we will directly multiply 5/2 by 3/4. Therefore in simple words 3/4 is multiplied by the multiplicative inverse of 2/5. When we multiply fractions we multiply straight across the top, 3 times 5 is 15 and 4 times 2 is 8. Thus 15/8 is the solution.

We also understand that in Division of Rational Numbers, we have to turn the fraction upside down and then multiply the first fraction by the resulted reciprocal. If we have two rational numbers 2 /3 and 3/4 and we need to divide ¾ by 2/3 so ¾ will be called as divisor, its reciprocal will be 4/3, which is nothing but done upside down. Then next step will be multiplying the first fraction with the reciprocated fraction. That is 2/3 multiplied by 4/3. Here we shall multiply 2 by 4 and divided by 3 multiplied by 3 and get 8/9 as the solution. We also will understand how we simplify the fractions, 48/108 we simplify by 2 the whole fraction as it is divisible by 2 we get 24/54. Another simplification by 2 gives us 12/27. Here we understand that 12 and 27 will not be simplified by 2 anymore as it’s not divisible by 2, thus which could be another number, 3 is the other number which is divisible so we get 4/9. Thus we simplify the fractions.