Sentences – Simple, Compound and Complex

Sentences can be of three different types – simple, complex and compound in English language. According to expert writers, writing should be mix of different types of sentences in order to be interesting and lively. Let’s have a look at each of the type along with examples in this post.

Simple Sentence: A simple sentence is also referred as independent clause. Simple sentences contain a subject and a verb that expresses the complete meaning and therefore, it is referred as independent clause.  For example:

1. Nappy Liners India has good collection of baby diapers.
2. Maria buys baby shoes online for her child.
3. She explored the online store to buy quality baby rompers online.

Compound Sentence: A compound sentence is a sentence that contains two independent clauses that are combined by a coordinator. Some of the popularly used coordinators are for, so, but, yet, and, or and so on. At times, coordinators in compound sentences are preceded by a comma. For example:

1. I bought nappies from Nappy Liners India online store and Mina bought baby shoes online.
2. John bought baby rompers online , so Jim planned to buy something else.
3. Tom went for shopping but didn’t buy anything.

Complex Sentence: A complex sentence is a sentence that has an independent clause joined by one or more dependent clause. A complex sentence always contains a subordinator. Popularly used examples of subordinators are: because, since, although, after and so on. For example:

1. Since she came to know about the online baby store, she has been buying baby products online.
2. Although I was not feeling well, I completed the task on time.
3. She came back home after she realized her mistake.

Identify simple, compound and complex sentence:

1. Mary likes dancing. (Simple Sentence)
2. I bought a cycle because I need to exercise a bit. (Complex Sentence)
3. I went for shopping and bought some interesting play toys. (Compound Sentence)
4. He is rich yet humble. (Compound Sentence)
5. She moved to the States after getting married with John. (Complex Sentence)

These are the basics about simple, compound and complex sentences.

Greater, Less or Equal

Greater, Less or Equal
Greater, less and equal is one of the most important and basic concepts of mathematics. Taught in the primary school, the concept gives a basic understanding of numbers and calculations. These three ideas are used while comparing two or more values. Let’s have a look at each of them in this post.

Greater Than and Less Than:
‘Greater than’ as the name suggests refers to a greater or larger value. While comparing two things, the greater is the one that has greater value.  On the other hand, ‘less than’ as the term defines refers to a lesser or smaller value. When two things are compared, the ‘less than’ refers to the one that has smaller value. Greater than and less than is represented by the greater than sign and less than sign i.e. “<” and “>”. The Big Value is > The Small Value. Similarly, The Small Value is < The Big Value. For example:

1. (5 tops for girls + 5 skirts for girls) > (3 tops for girls + 2 skirts for girls)
Here, the first value includes 5 tops and 5 skirts that is 10 clothing for girls while the second value includes 3 tops and 2 skirts that is 5 clothing for girls. Therefore, the first value as represented by the sign is greater than the second value.

2. (6 frocks for girls) < (12 frocks for girls)
Here, the first value includes 6 frocks while the second value includes 12 frocks and therefore, the first value as represented by the sign is smaller than the second value.
Equal to:
Besides greater than and less than, two values can also be equal. When two values are equal, they are represented by the equal sign that is “=”. For example:

3. Anita bought (2 baby brush, 2 baby soap, 1 baby powder, 3 baby diapers) online = Anita bought 8 baby products online.
Here, the first value sums up to buying 8 baby products online which is equal to the second value and therefore, the first value and second value as represented by the equal sign are equal.
These are the basics about greater than, less than and equal to concept.

All about Remainder Theorem

The remainder theorem is a useful method to evaluate the polynomials with a given value of ‘x’. Understanding this theorem on how to use will be more important while evaluating polynomial equations. This theorem usually starts with an unknown polynomial p(x), which means that a polynomial ‘p’ whose variable is ‘x’. Then this theorem is concerned with dividing that polynomial with some linear factor such as ‘x-a’, where ‘a’ can be any number. Then after dividing, we will end up with a polynomial answer like q(x) and some polynomial reminder r(x).

Remainder Theorem Definition
This theorem is based on the synthetic division, like the way of dividing the polynomial f(x) by another polynomial D(x) and then to find the remainder. This can be written in the form of f(x) = Q(x)* D(x) + R(x), where the dividend is f(x), quotient is Q(x), divisor is D(x) and the remainder is R(x). But this theorem steps forward and states that when a polynomial is divided by a linear divisor ‘x-a’, then the remainder will be equal to f (a).

Remainder Theorem Proof
Here we have to prove that the remainder of f(x) when divided by ‘x-a’ will be equal to the value of f (a). Let us consider a polynomial p(x) which is divided by ‘x-b’. Consider that q(x) will be the quotient and R as the remainder. Now, by the division algorithm we know that Dividend will be equal to divisor multiplied by quotient which is summed up with remainder. Hence we will get p(x) = q(x)*(x-b) + R. Now substituting ‘x’ value with b, we will get p (b) = q (b)*(b-b) + R and then we will get p (b) = R (b-b=0, 0-q (b) =0). Hence finally we will get the remainder as p (b) from this equation.

Remainder Theorem Examples
Let us take an example of a polynomial equation(x) = x3-7x-6, which is to be divided by a linear factor of ‘x-4’. Now when we commonly divide this polynomial by normal synthetic division, we will get the answer of remainder as 30. But now as per Remainder theorem, this value will be equal to value of f (4). Now substituting the 4 in f(x), we will get f (4) = (4)3-7(4)-6 whose value is also 30. This final step can also done by another method as writing in a table of coefficients as 1, 0,-7 and -6 and then placing 4 on the left side and multiplying with the remainder of the subtraction with the above numbers to the next number. By doing this so, the final remainder value will give the remainder.

There is Remainder theorem solver available on many websites such as http://calculator.tutorvista.com/remainder-theorem-calculator.html  in which by entering the equations it will automatically calculate the remainder and derive it.

Properties of Equality

Properties of equality are one of the major concepts in basic mathematics. Before realizing the properties of equality, let’s have a revision of the basic operations of mathematics – addition, subtraction, multiplication and division. Addition refers to combination of objects resulting larger number. Subtraction refers to reducing a part from a collection. Multiplication means scaling one number by another and division is that which includes a dividend, quotient, divisor and remainder. For example: Mother bought food for kids of Rs.500, nestle products for babies for Rs. 200 and some angry bird toys India for John for Rs.150. She paid total of 500 + 200 + 150 = Rs. 850 (addition). She bought three angry bird toys India for John and later gave one to Tina from the collection. John after that has 3-1 = 2 angry bird toys (subtraction). John said he will allow his five friends to play twice with the angry bird toys. Al together they played 5x2 = 10 times (multiplication). Mother bought 6 nestle products for two babies equally. Each baby has 6/2 = 3 nestle products (division).

Properties of equality are applied to all these four mathematical operations. There are eight properties of equality.
1. Reflexive Property: This refers to x=x. For example: 2=2, 3=3, 4=4 or I am equal t myself.
2. Symmetric Property: This refers to x =y, y =x. For example Cerelac food for Rs. 250 and Gerber food for Rs.150 = total food for kids costing Rs. 400.
3. Transitive Property: This refers to x=y, y=z, then x=z. For example: Ana’s weight is equal to Tina’s weight and Tina’s weight is equal to Peter’s weight. Therefore, Ana’s weight is equal to Peter’s weight.
4. Addition Property: If x=y, then x+ z = y+ z. For example: Ana’s weight is equal to Tina’s weight. Then, Ana’s weight + 3 kg = Tina’s weight + 3 kg.
5. Subtraction Property: If x= y, then x-z = y – z. For example: Ana’s weight – 2 kg = Tina’s weight – 2 kg.
6. Multiplication Property: If x= y, then x * z = y * z. For example: Ana’s weight * 2 kg = Tina’s weight * 2 kg.
7. Division Property: If x= y, then x/z = y/z. For example: Ana’s weight / 2 kg = Tina’s weight / 2 kg.
8. Substitution Property: If x = y, then y can be substituted for x in any mathematical expression.
These are the properties of equality.

Basic geometry definitions

Now a day’s tutor will teach the students through online. The tutors help the students the students to learn their doubts. The tutors gave neat and clear explanation. The geometry is the basic process in the mathematics. The measure of the angles, shapes is said to be geometry. The basic shapes in geometry are triangle, rectangle, square, and quadrilateral. In this article we learn about basic geometry definitions .

Basic Shapes – Basic Geometry Definitions:

Basic shapes in the geometry

Circle
Triangle
Rectangle
Square
Parallelogram
Trapezoid
Kite
Polygons

Definitions for the Basic Shapes in the Geometry:

Circle:

A circle is said to be a line segment. The starting point of the circle and the ending point of the circle is same. The shape circle is a 2 dimensional shape.

Triangle:

The shape triangle has three sides, three faces, and 3 vertices. This shape is also known as 3 dimensional shapes. There are various types of triangle

Types of triangle:

Right angled triangle: The triangle which has the accurate measure of 90 degree is said to be right angled triangle.

Acute triangle: The triangle which has less than 90 degree is said to be acute triangle.

Scalene triangle: The shape triangle has three angles, if all the three angles in the triangle are different is said to be scalene triangle.

Isosceles triangle:
The sides of the triangle which has two equal sides and one opposite side is said to be isosceles triangle.

Equilateral triangle:
The sides of the triangle which has all the three sides of the triangle are equal in measure is said to be equilateral triangle.

Rectangle:
The shape rectangle has four sides, four vertices, and four edges. This is said to be a 2 dimensional shape.

Mathematical Constant Pi

There are many constants in mathematics. The mathematicians are always fascinated by them. They try to find its value. Pi is one among them. We have been using it in our various calculations and are very important. If we take the circumference of a circle and divide it by its diameter, then we get the constant called Pi. The value of Pi is found to be approximately equal to 3.14159.Pi is sometimes represented by the symbol ∏. We have studied about the rational numbers. But the constant Pi is found to be irrational. It is sometimes represented by a fraction 22/7 for calculation purposes. But it does not represent the true value of Pi. It has several digits after the decimal point and they don’t repeat and also they don’t end. So, the digits of Pi after the decimal point go on. The history of Pi is quite interesting. The people found its existence in the 15th century.

Many great scientists and mathematicians have tried to find its value accurately. This process is an ongoing one. With the advent of technology and supercomputers its value has been found to several decimal places. Even values up to ten trillion places have been found out. The Pi value is also used in trigonometry. It is expressed in radians in trigonometry. Usually we measure angles in degrees, but in radians we use Pi. It is very important in the field of geometry as well. Many formula use Pi, without which many calculations are not possible. Even in aeronautical engineering and aerospace engineering calculations Pi plays a very important role.

So basically we need to understand what is the value of Pi to use it in calculations. It has infinite number of digits after the decimal point. It is not a rational number. It cannot be expressed as a number which can be written as P/Q. So, basically we can say Pi belongs to the set of irrational numbers. Finding the digits of Pi has going on over the years. But all those digits are not used by us in our calculations. Sometimes they are calculated by mathematicians to satisfy their curiosity. This process will go on. More results are coming out every day. We used to the use the value 3.14 of Pi in our calculations as it was very simple to use and the calculations would not become complex. This was done only to keep calculations simple.