Integers Rational Irrational Numbers

Introduction to integers rational irrational numbers:

Let us study about integers rational irrational numbers. Integer numbers are commonly seemed as the set of numbers that starts from infinite negative side to the infinite positive side.
Rational numbers are defined as numbers that can express themselves in the form of ratios as `m/n` where ‘n’ should not be equal to equal to ‘0’.
The numbers which cannot express themselves in the simple ratio are named as irrational numbers.

Integers Rational Irrational Numbers:

Example 1: Add the following numbers: 0.27, `3/2` , 1.52, -3, 2.5, 7.


Solution:

The given numbers are 0.27, `3/2` , 1.52, 3, 2.5, 7.
To add the given numbers the following steps are to be considered:
= 0.27 + `3/2` + 1.52 - 3 + 2.5 + 7

= 0.27 + 1.3 + 1.52 - 3 + 2.5 + 7 (do conversions of the fractions to decimal numbers)

= 9.59

Example 2: Subtract the following numbers by using same given order: 3.9, `4/5` , 0.13, 4, 1


Solution:

The given numbers are 3.9, `4/5` , 0.13, 4, 1.
To subtract the given numbers the following steps are to be considered:
= 3.9 – `4/5` – 0.13 – 4 – 1.

= 3.9 – 0.8 – 0.13 – 4 – 1. (do conversions of the fractions to decimal numbers)

= -2.03

Example 3: Do the multiplication process for the following numbers:  1.1, 0.4, 3, 9, `6/5` .


Solution:

The given numbers are 1.1, 0.4, 3, 9, `6/5` .
To add the given numbers the following steps are to be considered:
= 1.1 * 0.4 * 3 * 9 * `6/5` .

= 1.1 * 0.4 * 3 * 9 * 1.2 (do conversions of the fractions to decimal numbers)

= 14.256

Example 4:Do the division process for the numbers: `9/4.5` .


Solution:

= `9/4.5`
= 2


Exercises:

Add the following numbers: 0.2, 6, `11/5` , 4 (Answer: 12.4)
Subtract the following numbers by using same given order: 7, `5/8` , 1.3, 2. (Answer: 3.075)
Do the multiplication process for the following numbers:  0.09 0.3, `15/2` , 1, 7 (Answer: 1.4175)
Do the division process for the numbers: `2/0.5` (Answer: 4)

Adding Radicals Calculator

Introduction to adding radicals calculator:

Radicals:

The symbol used to indicate root of a number is known as radicals.  The square root, cube root can be expressed by using the radical symbol.  The radicals also mean that the root.  To understand the radicals, first we need to know about exponentiation. That is exponential xn = x * x * x * x …….n times x. The root of xn can be expressed as root (n)(x^n) = x .

The radical symbol is,

Fig(i) Radicals

Calculator:

A calculator is a small (often pocket-sized), usually inexpensive electronic device used to perform the basic operations of arithmetic.(source : Wikipedia).

Fig(ii) Adding radicals calculator

In this article we are going to learn about how to adding radicals using calculator.

Rules to Adding Radicals :

Before we get into problems , we need to know the rules for adding radicals.

Rules :

We can add the radicals with the same index and radicand . For example the radicals of index 2 can only add. We can add the radical of index 2 with the radical of index 3.

For example , `sqrt(4)` + `sqrt (4)` = ( 1 + 1) `sqrt(4)`

= 2 `sqrt(4)`

Let us see some problems on adding radicals by using calculator.

I am planning to write more post on Factorization Calculator, Equivalent Fractions Calculator. Keep checking my blog.

Problems on Adding Radicals Calculator:

Problem 1:

Add the radicals `5sqrt(36)` + `2sqrt(81) `

Solution:

Given, `5sqrt(36)` + `2sqrt(81) `

We need to add the given radicals.

Before that we can simplify the given radicals.

We know that, `root(n)(x^n) ` = x

`sqrt(36)` = `sqrt(6^2)` = 6

`sqrt(81)` = `sqrt(9^2) ` = 9

`5sqrt(36)` + `2sqrt(81)` = 5*6 + 2*9

= 30 + 18

= 48

Answer: 5`sqrt(36)` + 2`sqrt(81)` = 48

Problem 2:

Add the radicals `8sqrt(8)` + `sqrt(8) `

Solution:

Given, `8sqrt(8)` + `sqrt(8) `

We need to add the given radicals.

Before that we can simplify the given radicals.

We know that,` root(n)(x^n)` = x

`sqrt(8)` = `sqrt(4 * 2 )`

= `sqrt(4)` * `sqrt(2)`

= `sqrt(2^2)` * `sqrt(2)`

= `2sqrt(2)`

`sqrt(8)` = `2sqrt(2)`

8`sqrt(8)` + `sqrt(8)` = 8 * `(2sqrt(2)) ` + 2`sqrt(2)`

= 16 `sqrt(2)` + 2`sqrt(2)`

= (16 + 2)`sqrt(2)`

=  18`sqrt(2)`

Answer: 8`sqrt(8)` + ` sqrt(8) ` = 18`sqrt(2)`

Problem 3:

Add the radicals 3`root(3)(9)` + 2`root(2)(8) `

Solution:

Given,3`root(3)(9)` + 2`root(2)(8)`

We need to add the given radicals.

Here , given two radicals are has two different index as well as different radicand.

So we can add it further.

My Previous Blog :- http://online-math-problems-solved.blogspot.in/2012/10/solving-literal-equation.html

Polynomials in One Variable

Introduction to polynomials in one variable
Polynomials in one variable, here we need to understand each terms very clearly. Polynomial here means, a mathematical expression consisting of  a sum/subtraction/multiplication but not division of variables , constant sand exponents.One variable means  expression consisting of one letter. Polynomial with one term is called as a "Monomial", two terms as "Binomial" and three terms as "Trinomials". Terms can be separated by addition sign or subtraction sign only but not by multiplication sign.So we can say in otherwords that polynomial is a sum of monomials . And each monomial in polynomial is called as a Term.


Examples of Polynomials with One Variable
Let us see some examples to understand better the polynomials with one variable:

1. 4x

here we have one term only with single variable x

2. 2x^3 + 4x^2 - x

Here we have three term with one variable x.

3. 2x^2

Here we have one term with variable x and the exponent as 2.

Polynomial the word poly means "many" and the word nomial means "terms". so polynomial means many terms. But we are learning here polynomials with one variable.

In polynomial we have constants , variables and exponents. we need to understand these with examples.

Constants can be any number as 3 , 2 , -20 , 3/2 etc.

Variables can be any letter as x , y , z , a , b etc

Exponents will be always positive integers such as 2x^2 , 2a3, 4y5 etc

Algebra is widely used in day to day activities watch out for my forthcoming posts on what s an algebraic expression and what is an example of an algebraic expression. I am sure they will be helpful.

Degree of a Polynomial with One Variable

The largest exponent of the variable is called the degree of a polynomial with one variable.

Let us take an example to understand better:

1. 2x4 + 4x^3 - x^2 + 8

This is a polynomial with one variable with the highest degree as 4 which is also called as the highest power or exponent of the given variable.

2. 3a^2 - 5

This  is a polynomial with one variable with degree 2.

3. x

Here the degree is 1. A variable without a power is considered to be as 1 always.

Standard Form of a Polynomial in One Variable
Standard form means writing the polynomial in one variable with the highest degree first.

Let us take an example:

3x^3 - 2x^2 +5

here the polynomial in one variable is written on that order of degree( highest degree first).

It is not necessary to write it in standard form , but it helps us in doing simplification.

Let us take few more examples:

1.Add: 3x + 4x^2+5 and 8x^2 +2x

writing it in standard form

=(4x^2+3x+5) + (8x^2+2x)

add the like terms

=4x^2+8x^2+3x+2x+5

=12x^2+5x+5

2. subtract: (3a3+ 8a^2+3) -(2a^2- 3a3+8)

write it in standard form

=(3a3+8a^2+3)-(-3a3+2a^2+8)

Reverse the sign

=3a3+8a^2+3 +3a3-2a^2-8

now add/ subtract the like terms

=3a3+3a3+8a^2-2a^2+3-8

=6a3+6a^2-5

3. Multiply: 3x^2 by 2x^3

Multiply the constant and add the exponent of the variable

=6x5

Conclusion

Polynomials in one variable is easy to simplify.

If we add / multiply polynomials in one variable we get the results as a polynomial itself.

Always while adding polynomials simply add the like terms( variable with same degree) together.

In case of multiplication , multiply the constants first then the variable and then add the degree of the variable.

While doing the subtraction change the sign to reverse sign of each term you are subtracting.

Discrete Math Notes

Introduction to discrete math notes
Discrete Mathematics deals with a number of selected topics in Mathematics those are essential to the study of many Computer Science areas. Since it is very difficult to cover all the topics, only two topics, namely “Mathematical Logic” and “Groups” have been introduced. These topics will be very much helpful to the students in certain practical applications related to Computer Science. In this article we shall discuss about discrete math notes.

Discrete Mathematics Statements Notes

Logical statement or Proposition:

A statement or a proposition is a sentence which is either true or false but not both.

Simple statements:

A statement is said to be simple if it cannot be broken into two or more statements. All the statements in (a) and (b) of Example 1 are simple statements.

Compound statements:

If a statement is the combination of two or more simple statements, then it is said to be a compound statement.

Negation of a negation:

Negation of a negation of a statement is the statement itself. Equivalently we write ~ (~ p) = p

Conditional and bi-conditional statements:

In Mathematics, we frequently come across statements of the form “If p then q”. Such statements are called conditional statements or implications. They are denoted by p ? q, read as ‘p implies q’. The conditional p ? q is false only if p is true and q is false. Accordingly, if p is false then p ? q is true regardless of the truth value of q.

Tautologies:

If the last columns of  truth table have the value only T a statement is called a tautology, i.e., it is true for all logical possibilities. If the last column of its truth table have the value only F, i.e., it is false for all logical possibilities is called as statement contradiction

Discrete Mathematics Notes Example Problems
Example 1: Construct the truth table for ((~p) ? (~ q))

Truth table for ((~p) ? (~ q))

P          q         ~ p      ~ q      ((~p) ? (~ q))

T          T          F          F          F

T          F          F          T          T

F          T          T          F          T

F          F          T          T          T



Example 2: Show that ~ (p ? q) = (~ p) ? (~ q)

Solution:

Truth table for ~ (p ? q)

P          q         p ? q    ~ (p ? q)

T          T         T          F

T          F          T          F

F          T          T          F

F          F          F          T

Truth table for ((~ p) ? (~ q))

p          q          ~p       ~ q      ((~ p) ? (~ q))

T          T         F         F         F

T          F         F         T         F

F          T         T         F         F

F          F         T         T         T

The last columns are identical. ? ~ (p ? q) = ((~ p) ? (~ q))

Lowest Prime Number

Introduction :

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.                                                                                                                                                                                                                                        (Source:  Wikipedia)

Lowest Prime Number:

In prime numbers two is the very lowest prime number. It contains only two factors that is one and itself only. It has no third integer.

Between, if you have problem on these topics all the solve my math word problem, please browse expert math related websites for more help on prime numbers up to 100 chart.

Example Problems for Lowest Prime Number:

Problem 1:

Find out the number 31 prime number or not?

Solution:

The given number 31 is a prime number. Because it has no more values other than 1 and 31. Hence it is considered as a prime number.

Problem 2:

Find out the number 67 is prime number or not?

Solution:

The given number 61 is considered as a prime number. Because it is divisible by 1 and 67 only.

Problem 3:

Find out the number 117 is prime number or not?

Solution:

The given number 117 is not a prime number. The number divisible by more than two factors. Those factors are 1, 3, 9, 13, 39 and 117. Hence it is called not a prime number.

Problem 4:

Find out the number 217 is prime number or not?

Solution:

The given number is 217. So let us find that number is prime number or not.

The factors for the number 217 are 1, 7, 31 and 217. So the number contains more than two factors. Hence we cannot say it is a prime number.

Problem 5:

Find out the number 319 is prime number or not?

Solution:

The given number is 319. So let us find the given number is prime number or not. The prime factors for the given numbers are 1, 11, 29, and 319. Here the number 319 has more than two factors. So 319 is not a prime number.

Practice problem for lowest prime number:

Problem 1:

Find the number 53 is prime number or not?

Solution:

The getting number 53 is considered as a prime number.

Problem 2:

Find the number 89 is prime number or not?

Solution:

The getting number 89 is considered as a prime number.

Problem 3:

Find the number 78 is prime number or not?

Solution:

The given number 78 is not a prime number

Problem 4:

Find the number 110 is prime number or not?

Solution:

The given number 110 is not a prime number.

Prime Consecutive Numbers

Introduction to prime consecutive numbers:

A prime number is an number that has only two factors and is divisible only by 1 and itself, and by no other positive integer.The first few prime numbers are 2, 3, 5, 7, 11,13, 17, 19, 23, 29, 31 and 37.

The consecutive number is nothing but adjacent numbers for example the consecutive numbers of 2 is 3, even consecutive number between 1 to 10 is 2,4,6,8,10.

The prime consecutive number are of rare as the prime number were not of adjacent values normally. But there are two prime numbers that is also consecutive. the two and three are the only prime consecutive numbers. Let us see more about  prime consecutive numbers.

Discussion on Prime Consecutive Numbers

More about prime numbers:

A prime number is a natural number larger than 1 which cannot be expressed as the product of two smaller natural numbers.

Prime numbers were first studied rigorously by the ancient Greeks. One of the first theorems they proved was

Euclid’s theorem (˜ 300 BCE) There are infinitely many prime numbers.

Arithmetic progressions of primes

As we mentioned earlier, we are still unable to detect several types of patterns in the primes. However, we have made recent progress on one type of pattern, namely an arithmetic progression a, a + r, . . . , a + (k - 1)r.

In particular, for any given k, the primes contain infinitely many arithmetic progressions of length k. This result builds upon a number of existing results; for instance, in 1939, van der Corput showed that the primes contained infinitely many arithmetic progressions a, a + r, a + 2r of length three.

2
2, 3
3, 5, 7
5, 11, 17, 23
5, 11, 17, 23, 29
7, 37, 67, 97, 127, 157
7, 157, 307, 457, 607, 757
. . . . . . .  .  . ..  .  .  .  .   .   .   . .  . .
I am planning to write more post on find the prime factorization of 36, multiplying and simplifying rational expressions. Keep checking my blog.

More Discussion on Prime Consecutive Numbers

Let us take an example of number of prime numbers from the 1 to 1000 and try to find the number of prime consecutive numbers that can occur in that sequence.

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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997.

Here if we notice the above given prime numbers from 1 to 1000 numbers. The ones digit of the prime numbers are all odd numbers except the 2, also the 2 and 3 are the consecutive numbers and also prime numbers.

The 2 and 3 are the only prime consecutive numbers as all the other higher values are  with odd values, however these may satisfy the condition of prime consecutive odd numbers. the consecutive number in the list of prime numbers do not occur.

Significant Relationship

Introduction to significant relationship:

In mathematics, the word significant indicates significant figures or significant digits. There are some rules to identify the significant digits. In this section, we shall discuss the significant relationship. The following rules and solved example problems will help the students to prepare for significant relationship.

Significant Relationship :

The following points represent relationship between the uncertainty estimates and the significant digits.

Fining a number to 3 significant digits shows that the relative improbability or uncertainty in that number is less than 1%.

If we know a number to 6 significant digits, the relative improbability or uncertainty is less than 0.001 %.

Some definitions for significant relationship can be shown as follows:

Relative uncertainty = `max (|described Value - True Value|/|True Value|)`

Absolute uncertainty = `max (|described Value| - |True Value|)`

Rules for identifying significant figures:

Rule 1:

All non zero figures in any number are significant.

That is,   353.76 has five significant figures.

In addition, 76 has two significant figures.

Rule 2:

Zeroes between the other figures are also significant.

That is,   23009 has five significant figures.

In addition, 9.02 has two significant figures.

Rule 3:

Zeroes located at the last part of decimal values are also significant.

That is,   2.450 has four significant figures.

In addition, 7.0 has two significant figures.

Rule 4:

Zeroes located at the start of decimal values are not significant.

That is,   0.00093 has two significant figures.

In addition, 0067 has two significant figures.

Then 0.000008 has one significant figure.

Rule 5:

Zeroes located at the last part of the whole number are indefinite.

For example, 7800 has at least two significant figures.

However, it may also has more than two significant figures as shown as follows.

7.8 x 103    has two significant figures

7.8 x 103   has three significant figures

7.800 x 103 has four significant figures.

Algebra is widely used in day to day activities watch out for my forthcoming posts on algebra equation solver free and define an algebraic expression. I am sure they will be helpful.

Example Problem for Significant Relationship :

Identify the number of significant digits from the following.

56, 0.023, 7.2530, 7003, 2300

Solution:

56 – The number 56 is a non-zero number that has two significant digits.

0.023 – Zeroes placed before the other numbers are not significant digits and the number of significant digits in 0.023 is two.

7.2530– Zeroes placed after the other numbers but after a decimal point are significant digits and the number of significant digits in 7.2530 is five.

7003 – The number 7003 has four significant digits because the Zeroes placed between the digits are always significant.

2300:

The number 2300 may have at least two significant digits. It also has three and four significant digits.

That is:

2300 – 2.3 x 103 has two significant digits.

2300 – 2.30 x 103 has three significant digits.

2300 – 2.300 x 103 has four significant digits.