Scientific Notation Definition

Scientific Notations are standard way of writing number which are too small or too big.
In this standard notation all numbers are written in the form of:
m * 10n
here exponent n is an integer, m is mantissa which is a real number.
Scientific Notation Rules are as follows:
The value of mantissa holds a real number from 1 to 9 and exponent n contains the place value to get the original number.
If the decimal shifts to left by ‘n’, then the value of exponent increases by ‘n’. if decimal shifts to right the exponent decreases by 1 for each shift.
For addition or subtraction of numbers in scientific notations form their exponents should be same.
For multiplication of two such numbers, mantissa is multiplied and exponents are added.  
For division exponents are subtracted.

Scientific Notation Problems are as follows:
Q.1) Write the following numbers in scientific notations form:
1) 56788
2) 67.345
3) -6890
4) 0.000009888
5) 978.0 * 10-5
Sol.1)
1) 56788 = 5.6788 * 104. As we know that the mantissa can be real number from one to 9 so we shifted decimal to left four times and hence the exponent of 10 increases by 4.
2) 67.345 = (67.345)*100, in this number, value of mantissa is greater than 1 hence we shift the decimal to left and increase exponent by 1. So its scientific notations form is: 6.7345 * 101
3) -6890 = (-6.890)*103 by using rule used in above solution.
4) 0.000009888 = (0.000009888)*100. This number is less than one hence we will shift the decimal to the write and decrease the decimal by 1 for each right shift.
(9.888)* 10(0-6) = 9.888*10-6.
5) 978.0 * 10-5 is in exponential form but not in standard scientific notations as mantissa is greater than 1. So, shift it to let two time and increase exponent by 2 to get 9.78 * 10-5+2 = 9.78*10-3

Q.2) Solve the following:
1) (7.0  x  102)+(9.4 x 106)
2) (9.4 x 104)(3.5 x 10 –5)
3) (3.5  x  10 – 2)/(9.6 x 10 – 4)
Sol.2)
1) As exponents are not same. 9.4x106 = 94000 x 102
    (3.0  x  102)
  + (94000.0  x  102)
      94003.0 x 102 = 9.4003 x 106
2) (9.4 x 104) (3.5 x 10 –5)=(9.4 x 3.5) x 10(4+(-5))  (exponents are added as multiplication operation)
= 32.9 x 10-1
= 3.29 x 100
3) (3.5  x  10 – 2) / (1.5 x 10 – 4)
= (3.5/1.5) x 10 (-2-(-4)) (exponents are subtracted as division operation)
= 2.33 x 102

Present Tense and its types

Tense in English refers to the time of the verb’s action or state of being. There are three types of tenses – Present tense, Past tense and Future tense. Present tense refers to present action or state of being; past tense refers to past action or state of being while future tense refers to a future state of action or being. These three major types of tenses can be further classified into sub-categories. Let’s have a look at present tense and its classifications in this post.
Present Tense: Present tense is a type of tense that refers to some action or expression in the present time. For example: I am exploring online baby stores for new born shopping for my niece. Here, the sentence is referring to an action of exploring online baby stores for new born shopping in the present time and therefore, it is in present tense.

Types of Present Tense:
•         Present Continuous
•         Present Perfect
•         Present Perfect Continuous

Present Continuous Tense
Present continuous tense talks about an action that is still going on in the present time i.e.  continuing. For example: My cousin is using mustard seeds pillow for her baby. Here, mustard seeds pillow is still on use and therefore, the sentence is in present continuous tense.
Present Perfect Tense
Present perfect tense is that which speaks about something that began in the past and completes in the present. For example: John has solved the jigsaw puzzle. Here, John started solving the jigsaw puzzle in the past and finished it in the present and therefore, the sentence is in present perfect tense.
Present Perfect Continuous Tense
Present perfect tense is that which speaks about something that began in the past and is continuing in the present. For example: Jane has been buying baby essentials from online shopping baby stores from quite some time. Here, Jane started buying baby essentials from online shopping baby stores in the past and continuing in the present and therefore, the sentence is in present perfect continuous tense.
These are the basics about present tense and its types.

Subject and Predicate

Subject and Predicate are the most important concepts of English grammar. Introduced in the middle-school studies, subject and predicate plays an important role throughout the usage of English language. Every complete sentence conveying a meaning has two parts – the subject and the predicate. Let’s have a look at these two parts, its definitions and examples for better understanding.
Subject:
The subject is what or whom, the sentence is about. In simple terms, a person or thing that is discussed, described or dealt with in a sentence is called a subject of a sentence. For example:
• Mina has bought Pigeon products for babies. (Here, ‘Mina’ is the subject because the sentence talks about her buying Pigeon products for babies.
• Pigeon has wide range of products like baby diapers, shampoo, soap etc. (Here, ‘Pigeon’ is the subject as the sentence is talking about the brand.)
Predicate:
Predicate is a part of sentence that speaks something about the subject. It is a verb that states something about the subject. For example:
• John’s parents usually buy baby products India collection from online stores. (Here, ‘usually buy baby products India collection from online stores’ is the predicate of the sentence as it talks about the subject.)
• My uncle has recently started buying accessories from baby product online India stores. (Here, ‘has recently started buying accessories from baby product online India stores’ is the predicate of the sentence.)
More examples:
• I love my pet a lot. (‘I’ is the subject while ‘love my pet’ is the predicate)
• Hari is a busy person. (‘Hari’ is the subject while ‘is a busy person’ is the predicate)
• Online baby stores have made shopping easy for parents. (‘Online baby stores’ is the subject and ‘have made shopping easy for parents’ is the predicate.)
• Jack sings well. (‘Jack’ is the subject and ‘sings well’ is the predicate.)
• India is our motherland. (‘India’ is the subject and ‘is our motherland’ is the predicate.)

Introduction to parallel line extender:

Properties of parallel lines are the under the base of Euclid's parallel property. Two lines in a single plane that not even intersect or meet at any point then it is called as parallel lines. In other words Parallel lines are nothing but “A pair of lines in a plane which do not intersect or meet each other” then they are called parallel lines.

i.e. m1 = m2,

Where, m1 = slope of the first line.

m2 = slope of the second line.

Parallel Line Extender Problem:

Example for parallel line extender: Define the equation parallel to 4y + 4x = 8 and the line extender, with point (6.5,4).

Solution:

Given 4y - 4x = 8 and the point (6.5, 4)

To detect the perpendicular line, we have to find slope first.

For finding the slope, we need to change the given equation into slope intercept form.

4y + 4x = 8

Add 4x on both side,

4y + 4x = 8

- 4x = -4x

4y = -4x + 8

Divide by 4 on both sides,

y = (-x + 2)

The obtained equation is in the form, y = mx + b

So, the slope from the obtained equation m = -1

From generally we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = -1

So, m2 = -1

The line equation is,

(y - y1) = m(x - x1)

(y - (4)) = -1 (x - ( 6.5))

(y - (4)) = (-x + 6.5)

Y - 4 = - x + 6.5

Subtract 4 on both sides,

y = -x + 2.5

Answer: Thus, the parallel line extender is given through the line y = -x + 2.5

Example for Parallel Line Extender:

Example for parallel line extender: Define the equation parallel to 2y - 2x = 4 and the line extender is gets through the point (-2, 0.5).

Solution:

Given 2y - 2x = 4 and the point (4, 3)

To detect the perpendicular line, we have to find slope first.

For finding the slope, we need to change the given equation into slope intercept form.

2y - 2x = 4

Add 2x on both side,

2y - 2x = 4

+ 2x = +2x

2y = 2x + 4

Divide by 2 on both sides,

y = (x + 2)

The obtained equation is in the form, y = mx + b

So, the slope from the obtained equation m = 1

From generally we know that the slope of parallel lines are equal i.e. m1 = m2

Here, m1 = 1

So, m2 = 1

The line equation is,

(y - y1) = m(x - x1)

(y - (3)) = 1 (x - (4))

(y - (3)) = (x - 4)

Y - 3 = x - 4

Subtract 3 on both sides,

y = x - 1

Answer: Thus, the parallel line extender is used to find the line y = x - 1

Introduction to positive divisors

In math divisor is defined as “the number that you are going to divide by”. Divisor of a given integer can also be called as the divisor of a given integer.

Dividend /Divisor = Quotient.

Positive number/ Negative number = negative number

Negative number/Positive number = negative number

Positive/Positive  = Positive

Let us have an example 16/2 = 8 here 2 is a divisor.

16/4= 4 here 4 is a divisor.

16/8= 2 here 4 is a divisor.

In the above shown example the divisors are positive and also known as positive divisors.

The positive divisors when the dividend is 16 are 8, 4, and 2.

Example Problems for Positive Divisors:

Example problem 1:

Find the positive divisors of 28

Solution:

Positive Divisors of 28 are 1, 2, 4, 7, 14, and 28

Where 28 /1 = 28.

28 /2 = 14.

28 /4 = 7.

28 /7 = 4.

28 /14 = 2.

28 / 28 = 1

Hence for 28 we have 6  Positive divisors

Moreover 28 is a composite number

Example problem 2:

Find the positive divisors of 52

Solution

Positive   Divisors of 52 are 1, 2, 4, 13, 26, and 52.

Where 52/1=52.

52/2=26.

52/4=13.

52/13=4.

52/26=2.

52/52=1.

Hence for 52 we have 6   Positive divisors

Example problem 3:

Find the positive divisors of 30

Solution

Positive Divisors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.

Where 30/1=30

30/2= 15.

30/3=10.

30/5=4.

30/6=5.

30/10=3.

30/15=2.

30/30=1

Hence for 30 we have 8 Positive divisors

Example problem 4:

Find the positive divisors of 100

Solution

The positive divisors of  100 are  1,2,4,5,10,20,25,50,100

Where 100/1=100

100/2= 50.

100/4=25.

100/5=20.

100/10=10.

100/20=5.

100/25=4.

100/50=2.

100/100=1.

Hence for 100 we have 9 Positive divisors

Example problem 5:

Find the positive divisor of 31

Solution

Given that 31 is a prime number

All the prime numbers have only two divisors

The two divisors are 1 and the numbers itself

So the positive divisor of the number 31 is 1, 31 alone.

Finally the divisors of the prime numbers are 1 and the number itself.

Practice Problems for Positive Divisors:

Problem 1:

Find the positive divisor of 63

Solution:

The divisors of 63 is 1, 3,7,9,21,63

So the number 63 has 6 divisors.

Problem 2:

Find the positive divisor of 37

Solution:

The divisors of 37 are 1, 37

Problem 3:

Find the positive divisor of 15

Solution:

The divisors of 15 is 1, 3, 5, 15

Problem 4:

Find the positive divisor of 35

Solution:

The divisors of 35 is 1, 5, 7, 35

Understanding how to graph parabolas

A parabola (math) in co-ordinate geometry is defined as the locus of a point whose distance from a fixed point in the plane and perpendicular distance from a fixed line in the plane (not passing through the fixed point) are equal. The fixed point is called the focus of the parabolic and the fixed line is called the directrix of the parabolic.

These have many applications. Parabola help us to understand the trajectory of a projectile. The reflectors in car head lights or speakers in a sound system or mirrors in a telescope are all parabolic in shape. Parabolic mirrors are also used to harness solar energy. These are only a few examples where parabolas are useful. Parabolic forms find application in various other sciences field as well. Therefore it is of prime importance that we know how to graph a parabola.

Graph the parabola:

Parabola graphs are of two types: (1) horizontal and vertical. The horizontal form is again sub categorized into two types: (a) the ones that open left and (b) the ones that open right. Similarly the vertical forms are also of two types: (a) ones that open up and (b) the ones that open down.

The general equation of a horizontal form is: x = a(y-k)^2 + h, where (h,k) is the vertex of the parabolic. Whether the parabolic form opens left or right would depend on the sign of the term ‘a’. If a is positive, then that opens right and if a is negative the same opens left.

The general equation of a vertical parabolic form is y = a(x-h)^2 + k, where again (h,k) is the vertex of the same. If the parabolic form opens up then the ‘a’ in the above equation would be positive. If ‘a’ in the above equation is negative then the parabolic form would open down.

As we can see from the equation of the para-bola stated above, these equations are quadratic. The standard form of a quadratic equation in x is f(x) = ax^2 + bx + c. This would be a vertical parabolic form. If we were to find the vertex from this equation it would be: h = -b/2a and k = f(h). There to if ‘a’ is negative, these open down and if ‘a’ is positive, the same opens up.

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.