Define Positive Integer

The definition of positive integer is one of the most important topic in mathematics. Positive integer is always present after the zero in the number line. The symbol used for representing the positive integer is called as the " + ". All the mathematical operations are done using the positive integer. In this article, we are going to see about the positive integer with brief explanation and some example problems.

Explanation to Define Positive Integer
The explanation for to define the positive integer is given below the following,

Addition problem for defining the positive integer
Subtraction problem for defining the positive integer
Multiplication problem for defining the positive integer
Division problem for defining the positive integer

Example Problems to Define Positive Integer

Example 1: Add the following positive integer, 36 and 24

Solution:

36 and 24

3 6

2 4

___

6 0

___

Result: Positive integer addition = 6 0

Thus, this is the result to define the addition problem problem of positive integer.

Example 2: Subtract the following to define positive integer, 36 and 24

Solution:

36 and 24

3 6

2 4

___

1 2

___

Result: Positive integer subtraction = 1 2

Thus, this is the result to define the subtraction problem of positive integer.

Example 3: Multiply the following to define positive integer, 36 and 24

Solution:

36 and 24

3 6

2 4

____

8 6 4

____

Result: Positive integer multiplication = 6 0

Thus, this is the result to define the multiplication problem of positive integer.


Example 4: Divide the following to define positive integer, 48 and 6

Solution:

48 and 6

6 ) 4 8 ( 8

4 8

____

0

____

Result: Positive integer division = 6 0

Thus, this is the result to define the division problem of positive integer.
Practice Problems to Define Positive Integer

Example 1: Add the following positive integer, 42 and 14.

Answer: 56

Example 2: Subtract the following positive integer, 42 and 14.

Answer: 28

Example 3: Multiply the following  positive integer, 42 and 14.

Answer: 588

Example 4: Divide the following positive integer, 56 and 7.

Answer: 6

Probability Distribution Definition

The word probability and chance are familiar to everyone. Many a time we come across statements like “It is possible that our school students may get state ranks in forthcoming public examination.

“Probably it may rain today”

Definition: The word chance, possible, probably, likely etc. convey some sense of uncertainty about the occurrence of some events. Our entire world is filled with uncertainty. We make decisions affected by uncertainty virtually every day. In order to think about and measure uncertainty, we turn to a branch of mathematics is called as probability.

Probability Distribution Definition-classical Definitions

Definition: If there are n exhaustive, likewise exclusive and in the same way likely outcomes of an experiment and m of them are favorable to an event A, and then the mathematical probability of A is defined as the ratio m/n.

Definition for random variable:

The outcomes of an experiment are represented by a random variable if these outcomes are numerical or if real numbers can be assigned to them. For example, in a die rolling experiment, the corresponding random variable is represented by the set of outcomes {1, 2, 3, 4, 5, 6} ; while in the coin tossing experiment the outcomes head (H) or tail (T) can be represented as a random variable by assuming 0 to T and 1 to H.

Types of Random variables:

(1) Discrete Random variable
(2) Continuous Random variable


Definition for Discrete Random Variable: If a random variable takes only a finite or a countable number of values, it is called a discrete random variable.

Example:
1. The number of heads obtained when two coins are tossed is a discrete random variable as X assumes the values 0, 1 or 2  which form a countable set.
2. Number of Aces when ten cards are drawn from a well shuffled pack of 52 cards.
Probability Distribution Definition-theoretical Distributions:

Theoretical probability distributions is classified into

1. Binomial Distribution

2. Poisson Distribution

3. Normal Distribution

4. Exponential Distribution

Multiplication Of Exponents

Exponents are significant in scientific notation, when large or small quantities are denoted as powers of 10. exponents are mentioned by superscripts, as in the examples above. But it is not always possible way to write them this method.  If x is the exponent to which is a minimum base quantity a is increased value, then a x can be written in ASCII as a power of x. In a scientific notation, the higher case letter E can be used to point out that a number is raised up to a positive or negative power of 10. For model take 125x3. Here 125 is coefficient of variable 'x’ Then 3 is the exponent value of x . Exponent value also known as power value.
Suitable Examples for Multiply Exponents


Exponents of 1 and 0
If the exponent is 1, then only have the variable itself (example a1 =a)

Generally need not to write the "1", but it sometimes helps to remember that y is also a1
Exponent of  0

If the exponent is 0, then the values are not multiplying by anything and the result is just "1" (example a0 = 1)
Multiplying Variables with Exponents

multiply this variable with exponents:

(z2)(z3)

So that z2 = zz, and z3 = zzz so that all the multiplies,

(z2)(z3)

= zzzzz

That is 5 "z"s multiplied mutually so the new exponent must be 5:

(z2)(z3)

= z5

The exponents say to that there are two "z"s multiplied by 3 "z"s for a total of 5 "z"s:

(z2)(z3)



= z2+3 =z5

So, the simplest method is to just add the components
Mixed Variables for Multiply Exponents:

Have a mix of variables:

Example 1:

=xy2z y3z

=x y2+3z1+1

=xy5z2

so the result is =x3y5z2    

Example 2:

=x3y3z3   x2yz

=x3+2 y3+1 z3+1

=x5y4z4

so the result is =x5y4z4

With constant examples:

Example 1:

=5xyz 4xyz

=(5 4)x1+1 y1+1 z1+1

=20x2y2z2

The result is =20x2y2z2

Example 2:

=7xyz 3 x2yz

=(73 ) x1+2y1+1z1+1

=21x3 y2z2

The result is  =21x3 y2z2    

Meaning Of Prime Numbers

In mathematics, a prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The first twenty-five prime numbers are: Prime Numbers:2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.Condition: - If x is the prime number then the next factors of the number x is 1 and X. Let us see meaning of prime numbers.(Source:Wikipidea)                                                                                                                                                                                                                                                                              

Meaning of Prime Numbers:

Prime number means that number should divide by one and the number itself only. Otherwise it is not a prime number. If any one number divisible by two means that is not a prime number.

            Let us take a number to find prime number or not. For that we will take 131.

Step 1:

            Let us find the square root for that given number 131 = 11.44
            Now we have to change the decimal number as a whole number. That should be the nearest big whole number. 11.44 = 12

Step 2:

            Find the prime numbers below 12. The prime numbers up to 12 are 2, 3, 5, 7 and 11.

Step 3:
            So from this factors no one number will not divide by 131. The number contains 1 and 131 only.  Hence it is considered as a prime number.
Example Problems - Meaning of Prime Numbers:

Problem 1:

            Find out the number 29 is prime number or not?

Solution:

            Here the number 29 is divisible by one and itself only. It has no more factors other than this. So 29 is considered as a prime number.

Problem 2:

            Find out the number 53 is prime number or not?

Solution:

            The getting number 53 is not divisible by two. 53 has only two factors. Those factors are one and itself only. So we can say the given number is prime number.

Problem 3:

            Find out the number 78 is prime number or not?

Solution:

            The obtain number 78 is not a prime number. 78 is divisible by two. The number 78 contains more than two factors. Those factors are 2 x 3 x 13. Therefore the given number 78 is not a prime number.

Problem 4:

            Find out the number 108 is prime number or not?

Solution:

            The obtain number 108 is not a prime number. Why means 108 is divisible by two. 108 includes more than two factors. Those prime factors are 2 x 2 x 3 x 3 x 3. Therefore we cannot say it is a prime number.
Practice problems - Meaning of prime numbers:

Problem 1:

            Find the number 13 is prime number or not?

Solution:

            The given number 13 is considered as a prime number.

Problem 2:

            Find out the number 54 is prime number or not?

Solution:

            The given number 54 is not a prime number.

Solving Polynomials

What are Polynomials?
The simple definition of Polynomial is that, it is an expression containing multiple terms that are combined together through addition, subtraction and multiplication. This polynomial definition is illustrated by this polynomial equation: 3x + 4y + 5. In this example, the polynomial equation is a combination of three terms.

Ways of Solving Polynomials
There are different ways of solving polynomials and the strategy differs based on the polynomial equation provided. In this article, let us discuss two different ways to solve polynomial through relevant examples.

Example 1: Solve polynomial (a+8) (2a-10)

In this equation, there are two sets of terms that are multiplied with each other. To solve this polynomial, multiply each term of the first set with all the terms in the second set. This is illustrated below:

When ‘a’ is multiplied with ‘2a’, it gives 2a2. Then ‘a’ is multiplied with -10, which results in ‘-10a’. As a next step, multiply 8 with the terms in the second set. When ‘+8’ is multiplied with ‘2a’, it gives ‘+16a’ and ‘+8’ multiplied with ‘-10’ gives ‘-80’. Now the equation is dissolved into:

(a+8) (2a-10) = 2a2 -10a + 16a -80

In the above equation, the second term (-10a) and the third term (+16a) contains the same term with the same degree, but the constant value alone varies. When there are several terms with same variable and same degree, they can be combined. In this case, -10a+16a can be rearranged as 16a-10a, which when solved results in 6a. Thus, the polynomial equation is further solved into the result 2a2 + 6a – 80.

Example 2: Solve polynomial equation a+2b = 5a+7b
In this equation, there are two polynomials, one on the left hand side and the other on the right hand side. In this case, these polynomials help each other to find the solution. The polynomial solver for this expression will be evolved through the steps below:
• Original Expression: a+2b = 5a+7b
• Moving right hand side expression to left: a +2b-5a-7b=0.
• Grouping terms of same variables and degree: a-5a+2b-7b=0
• Combining terms of same variables and degree: -4a-5b
• Take common factor out: -1(4a+5b)

Thus the result is -1(4a+5b).

When the polynomials are complex, the above strategies alone might not work. For instance, if you are solving quadratic polynomials then you have to perform series of steps such as finding zeros of the polynomial, finding roots and much more to solve them.

What is a Perfect Number?

A Perfect Number is nothing but a whole number, which is identical to the sum of its each and every proper divisor.

Perfect numbers

What are Divisors?
Divisors are same as the factors of a number.  A divisor or factor is a number which divides a number evenly and gives the remainder as zero. Finding factors of a number is simple. For example, consider the number 9.  The divisors or factors of 9 are 1, 3, 9, because only these three numbers can divide 9 without leaving any remainder.

Mathematical problems are solved by finding factors and multiples as we did now. In the above case, 3 is the factor of 9 as it divides 9 evenly. At the same time, 9 is a multiple of 3 because 3 X 3 = 9. This signifies that finding factors and multiples are necessary to find the perfect numbers and to solve many algebraic expressions.

What are Proper Divisors?
Proper divisors are the divisors of the number excluding the number itself. As already discussed, the divisors or factors of 9 are 1, 3 and 9. The proper divisors of number 9 are 1, 3, and 9. Number 9 is not a proper divisor of 9 because it is the same as the original number for which the divisors are identified.

What are Perfect Numbers?
Knowing what proper divisors are, now let us look at the question: What is a perfect numbers? Identify the proper divisors of a number and add them. If the result of addition is same as the actual number for which the divisors are identified, then that number is a perfect number.

For example, we identified 1 and 3 as proper divisors of9. Now add these proper divisors:
1 + 3 = 4

Here the sum is 4. Instead, if the sum was 9 (the actual number for which you found the divisors), then 9 will be called as perfect number. In this case, sum is 4 and not 9. This signifies that 9 is not a perfect number.

Let us take up another example. Are 6 a perfect number? The divisors or factors of 6 are 1, 2, 3, and 6. The proper divisors of number 6 will then be 1, 2 and 3. Now, add these proper divisors.  Addition of 1, 2 and 3 will result in 6. The result 6 is same as the number 6 for which we identified these proper divisors. Thus, 6 is a perfect number.

The other perfect numbers are:
• 28
• 496
• 8128

Applied Statistics learning

Let's learn about what is applied statistics in today's post.

Applied statistics is that section of statistics which can be applied in real life. There are different levels of measurements and these are:

• Nominal measurement
• Ordinal measurement
• Ratio measurement
• Interval measurement

Next time will help you learning on examples of analyzing data.

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