Statistics in IX Grade IX Grade

Hello Friends, in today's class we all are going to discuss about one of the most interesting and a bit complex topic of mathematics, Probability and statistics. Here I am going to tell you the best way of understanding probability problems and also discuss about its applications not just in mathematics world but also in other areas of studies as well. The other areas of study where it plays an important role are Physics, Chemistry, social science etc. In ninth grade what students suppose to do with probability and statistics are : They use an appropriate language to express their findings and duplications. Teachers tells them how to create questions to help them find the differences among several samples in a population. They assemble their studies of situations to include the part of experimental and random surveys. Students try to learn things to use and explain the univariate and bivariate in measurement and definitive data. The overall information is basically used to develop scatter plots, regression coefficients, and regression equations using technological tools.

Additionally, students also study the application of sample statistics for creating explanations using desired data analysis. This analysis is basically used to understand basic patterns of randomness for that probability that certain events may be independent of other events. Students need to understand the use of simulations to explain randomness of events.

Now Start with basic terminologies behind probability and statistics :
Probability and Statistics are used to understand the randomness and to collect, organize, describe, and analyze numerical data. Probability and statistics are dependent on each other starting from weather reports to studies of genetics, from election results to product preference surveys, probability and statistical language and concepts are increasingly present in the media and in everyday conversations. Students need this area of study to help them judge the correctness of a bickering supported by seemingly actuating data.

Firstly discuss about Probability : It is the study of random events. It is a way of telling or expressing a knowledge that an event will occur or has occurred. Probability theory plays an important role in various human activities that involves the analysis of a large set of data.

The probability of an event occurring given that another event has already occurred is called a conditional probability. The Mathematical view of this statement comes with the following equation that an event A occurs, given that an event B is already occurred.
P(B|A) = P(A and B) / P(A)
P(A/B) = Probability of occurrence of event A when the event B as already occurred and P(B/A) = Probability of occurrence of event B when the event A has already occurred.

The another method to calculate conditional probability is Baye's formula. It states that the probability of event B is the sum of the conditional probabilities of event B given that event A has or has not occurred. Mathematical expression for the above is : P(B) = P(B|A)P(A) + P(B|A^c)P(A^c)
And for the two independent events (for event A and event B) the expression is:
P(B)P(A) + P(B)P(A^c) = P(B)(P(A) + P(A^c)) = P(B)(1) = P(B)

Now let's take the basic behind statistics : Statistics is the practice or science of collecting and analyzing numerical data in large quantities. It is basically a study of the collection, organization, analysis and interpretation of the data. It is used to describe and analyze sets of test scores, election results, etc. Probability and Statistics together play an important role in finding out measures of central value, measures of spread, and helps in comparing of two data. These two are closely related to each other as statistical analysis or data are regularly analyzed to understand whether results can be formulated accurately about a particular event and also to make predictions on future or upcoming events. For example in an election time the early election results are analyzed to see if the results which is predicted is correct or not. They also try to predict or assumed the final outcome of the election.

So we can say that Statistics is the science which deals with variations , randomness and chance. It is different from other science as others works or studied on exact deterministic mathematics laws. In a lot of statistical analysis and experiments, the result depends on probability distributions as probability plays an important part in statistical analysis. Statistical analysis uses probabilities and probability calculations uses statistical analysis. For example in an experiment in Social Science, we assume a normal or common distributions for sample and population. The normal distribution is one of the probability distribution.

Let's take a basic example of probability theory to understand the basic concept:
Problem : Two coins tossed possibly multiple times and outcome is ordered pair:
Sample Space : (H, H), (T, H), (H, T), (T, T)
Field of subsets of all sample spaces : ø, (H, H), (T, H), (H, T), (T, T), sample space, (H, H), (T, H), ….
Solution : Let
A = (H, H), (T, H)
B = (H, T), (T, T)
Then:
P(A) = ½ , P(B) = ½
P(A intersection B) = ¼, P(A union B) = ¾


Mean and Mode :
The sample mean is basically an average and is formulated as the total of all the predicted outcomes from the sample divide by the total number of given events.
Let's take an Example to understand it better :
Problem : Suppose a randomly sampled six plots in the badland for a non-domestic weed and came up with the following counts of this weed in this region :
34, 43, 81, 106, 106 and 115

What we need to do is to compute the sample mean by adding and dividing by the number of samples by total number that is 6.

Solution:

34 + 43 + 81 + 106 + 106 + 115 / 6 = 80.83

Here we can say that the sample mean of non-indigenous weed is 80.83.

Mode : In a simple mathematical manner we can say that the mode of a set of data is the number with the highest frequency. In the above given example 106 is the mode. As in the above example 106 occurs twice and the rest of the possible outcomes occur only once.

The population mean is the average of the entire population and in real life it is impossible to compute.

Median

The problem with mean is that it mostly does not depict the original outcome. If only one possible outcome that is very much far from the rest of the data, then the mean will be affected by this outcome. This outcome is known as outlier. To avoid this situation we can use median. The median is basically a middle score. In this if we have an even number of events we can take the average of the two middle values. It is the best way for describing the typical value. It is mostly used for income and home prices.

Let's take an example to understand it better:

Problem: A person randomly selected 10 house prices in the city area. In Rs. 100,000 the prices were:

2.7, 2.9, 3.1, 3.4, 3.7, 4.1, 4.3, 4.7, 4.7, 40.8
If we are going to formulate the mean we would say that the average house price is Rs. 744,000.

3.7 + 4.1 / 2 = 3.9
The median house price is 390,000. This better reflects what house shoppers should expect to spend.

Another Example to understand it better : These are the given values, we need to apply both mean and median :

44, 50, 38, 96, 42, 47, 40, 39, 46, 50.

 

To find the sample mean, add them and divide by 10:
44 + 50 + 38 + 96 + 42 + 47 + 40 + 39 + 46 + 50 / 10 = 49.2
Noticeable thing is that the mean value is not a value of the sample.
Now on same set of data we need to find median:

To find the median, first sort the data:

38, 39, 40, 42, 44, 46, 47, 50, 50, 96
Notice that there are two middle values 44 and 46. To find the median we take the average of the two.
Median : 44 + 46 / 2 = 45.
Here the mean is larger than all. The mean is affected by outliers while the median is robust.

Law of averages:

Median : The Median is the middle value in the given data list. When the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order. When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two.

 

This is all for today. In next class I am going to discuss about other topics like variations, standard deviations etc.

IX grade Powers and Radicals

Friends today we all are going to learn one of the most common and important topic of IX grade mathematics which requires lot of practice and focus. Powers and Radicals are the two topics which we are going to discuss, as both topics come in almost each and every area of mathematics studies. Sometimes problem becomes so tough to solve as it might come with powers or in radical form. A daily practice and hard work will help you to get some confidence. Let's start with powers and will talk radicals later. It offers an admirable and compact explanation of two important principles in mathematics :

1. The first concept is to introduce concepts in a simple manner and then generalize them in such a way that rules and facts that are true in the basic manner remain true in the more general context.
2. The another concept is to reduce your problem to one you have already solved before.
   This is the way through which we can obtain powerful and efficient problem solving tecjniques and we can further improve our understanding of mathematics.

Now the topic to study is Multiplication : While doing multiplication we observe that it is defined by repeated addition. Thus for any number “x”

3 * x = x + x + x.

In a more pronounced manner, for any number “x” and any of the natural number n (that is one of 1,2,3,....) we define the product

n * x = x + x+ …..+x

here X is repeated n times.

Now we can extend this definition to factors n that are not natural numbers such as negative numbers, fractions, or, in general, real numbers etc. in such a manner that the ordinary rules of arithmetic always remain true.

The next topic to discuss is Natural Number Exponents

Powers work exactly the same manner except that instead of addition we use multiplication as our basic repeated operations. Thus, for any of the number x and any natural number n.

xⁿ = x * x * …...* x

Here n factors x

Lets take an example to understand:
2³ = 2 x 2 x 2 = 8
3² = 3 x 3 = 9
1ⁿ = 1 x 1 x …..x 1 = 1
0ⁿ = 0 x 0 x 0 x 0 = 0
x¹ = x
The expression Xⁿ is called a power and described in words as x to the power n. The number x is the base of the power and the number n is its exponent.

Now the next topic is the Central Rule :

2³ x 2⁴ = (2 x 2 x 2) x (2 x 2 x 2 x 2) = 2^{3 + 4} = 2⁷

For any real number x and natural numbers m and n

x^m x xⁿ = x^m+n

The above formula is called as central rule. We write down m factors x and then n factors x , and so we need to write down a total of m + n factors x.

Next topic is Zero Exponents

Now we are going to understand the case where the exponent is having a value zero. You should concentrate a bit more on this paragraph. It is elementary and easy to understand, and yet it shows a broad folder of mathematical thinking. Here we need to apply our central rule. Let's take an example :

2⁰ x 2³ = 2⁰⁺³ = 2³

 Here 2⁰ is basically a number that gives 2³ when multiplied with 2³. There is only one number that gives namely the number 1. So we are compelled to define 2⁰ = 1 . If a base was a number other than 2, say x, the same sort of argument applies, and so we define z⁰ = 1. There is one caution, however. If this x = 0 definition would give 0⁰ = 1. We also see that for any natural number n. we have that 0ⁿ = 0. If we were to expand that rule to n = 0 we would have 0ⁿ = 0. So we obtain different value of 0⁰, depending on which rule we want to apply. In this case, neither rule is better than the other, and so the most useful approach is to resign ourselves to the fact that 0⁰ is undefined. There is no problem when the base is non-zero however, and so we define

x⁰ = 1 (for all x not equal to 0)

Now I am going to teach the basic concept that is the exponent of a number says how many times to use the number in a multiplication.

Let's take an example to understand it better : 3² can be written as 3 x 3 = 9. 3² could be called as "3 to the second power", "3 to the power 2" or simply "3 squared".

Y = e^x The variable x in the equation is called the 'exponent', the function is called the exponential function.

The inverse of the exponential function is the logarithmic function or logarithm. This general logarithm is written as y = ^a log(x) or y = log _a (x).

The logarithmic function is defined for all positive real numbers x. So, the domain of the logarithmic function is the set of all positive real numbers. Operations as multiplying, dividing or raising to a power can be simplified to the adding and subtracting of logarithms by use or relations as log ab = log a + log b and log a^k = k log a.

Now the next topic is Scientific Notations : Scientific notation is a method developed by scientist. Scientists have developed a shorter method to express very large numbers. A number in scientific notation is written as the product of a number (integer or decimal) and a power of 10. This number is always 1 or more and less than 10. Scientific Notation is based on power of the base number 10.

The number 23,000,000,000 in scientific notation is written as:

2. 3. x 1011

here the first number 2.3 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called as the base value. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. The number 11 is referred to as the exponent or power of ten.

Following rules are applied when Scientific notation problems come into account:
Addition and Subtraction: All numbers are converted to the same power of 10, and the digit terms are added or subtracted.
Multiplication: The digit terms are multiplied in the normal way and the exponents are added.
Division: The digit terms are divided in the normal way and the exponents are subtracted.
Power of Exponentials: The digit term is raised to the indicated power and the exponent is multiplied by the number that indicates the power.

Roots of calculations : We need to change the exponent if necessary so that the number is divisible by the root.

Next topic we will study is radicals:

First question comes in our mind is that what is a Radical? Radical is known as root of an expression. A fraction is not a radical, but a fraction may contain a radical. It uses the the sign of radical (√) also known as surds. To understand it better let's take an example:

squares 2 = 4 it means 2 is the square root of four. 2³ = 8 this means 2 is the cube root of 8. if a, b are real numbers, n is a positive integer and if an = b then the n th root of b is a. then it can be written in this form :

n√b = a

n√b the radical sign, n is the indexed and b is the radicand. The index give the degree of the roots.

A radical equation is an equation in which at least one variable expression is stuck inside a radical, usually a square root. For example

√x + 2 = 6

2 + 2 = 4

( 2 + 2 )² = squared 4 here the final result shows that

16 = 16.

Various online radicals calculators are available which helps us to solve radicals. Simplifying Radical Calculator helps in getting an accurate answer in faster manner. It also provides the complete solution which helps us to solve radicals manually. By taking an Online Math help we can easily understand the basic concept behind the radicals equations or problems.

The steps to solve radical problems are: break or isolate the radicals to the left side of equal signs means get one radical on one side and everything else on the other using inverse operations. Then square each side of the equation and solve the equation we get all the solution.

Equation which contains radicals deals with square root equations whereas the square root symbol is known as radicals. Radicals also plays an important role in various engineering and physics branches. Various algebraic equations also come with the inclusion of radicals.

IX Grade Fractions

IX grade mathematics
Hello Friends today we all are going to understand the basic concept behind one of the most important and a bit complex of IX standard that is Fraction. Before proceeding further, let's talk about the basic concept behind Fractions. In simple mathematics a fraction is an expression that consists of numerator and denominator. For example a/b where a is the numerator and b is the denominator and both are integers. The most important condition is that denominator can never be a zero. Here we will try to understand some of the basic facts about fractions and also try to know the reason behind these facts.

Few of the facts about fractions are:

1. The integer “x” on the top of the fraction bar is the numerator and the integer “y” at the bottom is the denominator of the fraction.
   Fraction = numerator / denominator
   
2. Any of the integer can be considered as a fraction having a denominator equal to 1.
   X = X/1
   
3. If we are going to multiply numerator and denominator of a fraction with the same non zero factor does not manipulate the fraction. Basically in elementary school we classify it with the term equivalent fractions, but a better view is that equivalent fractions denote the same real number and need not be distinguished other than for simplicity.
   X/Y = CX/CY where C is not equal to zero.
   
4. A fraction of having the same denominator are added or subtracted by adding or subtracting the numerators.
   X/y + C/y = X + C / y
   X/y - C/y = X - C / y
   
5. If the fractions come with distinct denominators than firstly we need to turn them into fractions with the same denominators. This can be done by multiplying numerators and denominators with suitable factors, and then added using the rule above.
   X/y + c/d = Xd / yd + cy / yd = Xd + cy/ yd
   Similarly we can perform subtraction :
   X/y - c/d = Xd / yd - cy / yd = Xd - cy/ yd
   
6. In the case of multiplication, fractions are multiplied by multiplying numerators and multiplying denominators.
   X/Y * C/D = XC/YD
   
7. We can find the reciprocal of a fraction by switching numerators and denominators.
   
8. In division case : If we are going to divide a fraction by another fraction which is equivalent to multiplying with the reciprocal of the second fraction.
   X/Y divide C/D = XD / YC
   

Now lets go in detail with fractions. A fraction such as 5/4 is simplified in a kind of manner to explain the result of dividing 5 by 4. Commonly, A fraction X/Y explains the result of dividing X by Y. This explanation is true even if X and Y are not integers, an explanation of the principle that we make definitions in simple contexts and then generalize them so that all relevant rules remain true. However, if X and Y are not integer then the expression X/Y is called a ratio or quotient.
Division in a fraction can be defined as the solution of a multiplication problem.
Thus the fraction x = 5/4

is the solution of the equation
4x = 5
In more general form, the fraction
x = a/b

is the solution of the equation
bx = a.
The most important thing is to understand the basic definition as all of the above rules can be derived from it
Now let's see how to add fractions with different denominators:
Whenever we are going to add two fractions like 1/3 and 3/7 then we firstly need to convert them to fractions with the same denominator by applying the above rule number 3. Therefore we need to extract a common denominator. In a practical manner the more smaller the denominator, the more easier it is to manipulate the numbers and that time we mostly like to use the least common denominator. Meanwhile, the the product of the two denominators always works, and it is often, like in this case, also the least common denominator. So let's take this example to illustrate it in a more conventional manner:

1/3 = 1*7/3*7 = 7/21 and 3/7 = 3*3/7*3 = 9/21
Therefore:
1/3 + 3/7 = 7/21 + 9/21 = 7+9/21 = 16/21.


Now the next thing is how to multiply fraction :
Suppose we have two fractions that are x = 3/5 and y = 14/9
This shows that x and y satisfy the equations:
5x = 3 and 9y = 14
Now moving further and multiplying on both the sides of the first equation with 14 but we need to call 9y on the left hand side, we obtain
5x X 9y = 3 X 14
This can be further simplified as
45 xy = 42
which is the another way of explaining this:
xy = 42/45
Additionally we can further simplify this fraction by dividing numerator and denominator by 3 which gives us : 14/15


Now the next topic to illustrate is division of fractions:
Let's take the same above example and try to perform division operations with same fractions then:
a = x divide y = 3/5 divide 14/9
In this, division is the inverse procedure of multiplication, so it satisfies the equation
That shows as ay = x
or we can explain as
a X 14/9 = 3/5 We can find out the exact answer by multiplying on both sides of this equation with the reciprocal of y that is with 9/14
a X 14/9 X 9/14 = 3/5 X 9/14
This becomes : a * (14*9 / 9* 14) = 3*9/5*14
x divide y = a = 3*9 / 5*14 = 27/70.

 

Now the next topic to study is Greatest Common Factors and Least Common Multiples:
In simple mathematical manner a multiple of a natural number is the product of that number with another natural number. The most common example is, 6, 18, and 333 are all multiples of 3. Another thing to understand is that a factor of a natural number n is another natural number that divides n without remainder. Example to understand it, the factors of 12 are 1, 2, 3, 4, 6, 12. Similar concept is that a common factor of two natural numbers m and n is basically a number that is a factor of m and also of n. Let's take an example, 8 is a common factor of 32 and 48. Other common factors are 1, 2, 4, 8, 16.
Now we are going to understand the basic behind greatest common factor. The greatest common factor is just what the name suggests, so the greatest common factor of the above example that is 32 and 48 is 16. Similarly, a common multiple of two numbers is a multiple of both, and the least common multiple is the smallest common multiple. For example, 24, 120, 180 are all common multiples of 4 and 6, and 12 is the least common multiple.
The importance of this statement is that a common denominator of the two fractions is basically a common multiple of the two denominators and the least common denominator is generally the least common multiple of the two denominators.
The least common multiple (LCM) and the greatest common factor (GCF) of two numbers m and n are explained by the form:
LCM = m * n/ GCF
Let's take an example to understand is better, the Greatest common factor of 4 and 6 is 2 and their least common multiple is 12. So by using above formula we can formulate it:
12 = 4*6/2


Now the another area of study is Mixed Numbers or Mixed Fractions:
Mixed Numbers are basically fractions which are written as a natural number plus a fraction in which the denominator is always greater than the numerator. This area of study is important because the integer part in the fraction gives an indication of their sizes.
Mixed numbers are popular because the integer part gives an explanation of their size, but otherwise they have little to glorify them. They always form an exception (that is the only exception) to the rule that a unidentified or missing operator means multiplication, and they make the given arithmetic operations harder to carry out.


Now the another part of fractions are complex fractions. A complex fraction is a rational expression that has a fraction in its numerator, denominator or both. Example to show the complex fraction: a/b / c/d. The sign / denotes the division of two numbers or variables. Simplifying complex fractions involves following steps: firstly student needs to rewrite the numerator and denominator so that each form a single fraction. Now divide the numerator by the denominator by multiplying the numerator by the reciprocal of the denominator. In last step we need to simplify the rational expression.
So this is enough for today. In next class we all are going to learn the divisibility rules of the fractions and the use of Euclidean Algorithm while solving fractions.

Introduction to IX Grade Algebra

Friends today we all are going to rewind our basic concepts of Algebra along with ninth standard Algebra problems. We all know about the basic concepts behind numbers and how can we combine them with the help of basic operations of addition, multiplication, subtraction and division. This mathematics area of study is known as Arithmetic. In ninth standard we are going to study the more advanced area of Algebra which is distinct from arithmetic. In this section of Algebra we are going to perform basic operations for example addition to specific numbers that involves entities called as variables. Variables are those entities which have no particular value or an unknown value. The variables in mathematics are usually denoted by the upper or lower case letters. Algebra is a fundamental branch of mathematics that mainly deals with the rules of operations and their relations and the constructions and concepts arising from them. If we talk in simple words then we will see that algebra is simply the art of replacing variables in place of numbers.

Now I am going to discuss it in deep. An algebraic expression is basically a collection of numbers and letters combined by the four (addition, multiplication, subtraction and division) basic arithmetic operations. Here are some of the examples of algebraic expressions to understand it better.

7y, 3a+y, 3a-4y, a/(a+y), a², (a+b)²
The numbers used in the algebraic expression are called constants. Additionally, if there is no variable in the algebraic expression then we called it an arithmetic expression. For example 4 + 3 / 8.

Variables in the algebraic expressions are basically used to describe general conditions or situations or we can say that it models the real problems in the form of algebraic expressions and they can be used to solve problems that in other manner would be much more difficult or complicated or even impossible. These applications are used when we all are going to solve word problems.

Now moving forward let’s discuss about one of the most important and frequently used terminology that is an equation. An equation is basically an allegation that two algebraic expressions are equal or similar. This statement can be represented in two different ways or we can say that it can have two distinct meanings:

The first statement says that the equation is true for all values of the variables. In such kind of situations the equation is called an identity. An example of an identity to understand the above statement is
x + y = y + x
for all numbers x and y. This statement or example is called the commutative law of addition. Another well known identity is the first binomial formula which can be stated as :
 (a+b)² = a= + 2ab + b²
The second statement says that the equation is true for some of the values of the variables. Here we need to find out the values of the variables for which the equation is true. This terminology is called as solving the equation. Let’s take an example to understand it better. The equation to solve is: 3y + 1 = 4
in this equation we know that
y = 1
is the solution. The above equation is an example of a linear equation. Let’s take a bit more complicated example of a quadratic equation is y² - y - 2 = 0. Now after solving this we found that this equation has the two solutions
  y = -1    or   y = 2.
Now in Ninth standard we are also going to understand the basic concept behind evaluating an algebraic expression. To evaluate an algebraic expression can be stated as, substituting specific values for its variables. Lets’ take a very simple example to understand it:
Expression = 2y + 1
Evaluating expression, for the value of variable y = 3
give expression = 2 * 3 + 1= 7
We can say that value of expression at y = 3 is 7.
Let’s take an another way to solve which is a bit more complicated:
Expression at y = 2x + 1, where x is another variable
Expression = 2 * (2x + 1) + 1 = 4x + 2 + 1 = 4x + 3.

The next condition is if there are equivalent expressions:

Two expressions are equivalent if their values are equal for all possible estimations of the two expressions. In a simple mathematical manner we can say that listing expressions with an equality sign between them always gives an identity.

Now the next topic of the Algebra section is Number system. This section tells about the different types of numbers. Here we are going to talk about the four types of numbers that are Natural numbers, the Integers, rational numbers and real numbers.
Natural Numbers: 1, 2, 3, 4, 5, ..... are the natural numbers. We can add or multiply the two given natural numbers and obtain another natural number. Some of the mathematical laws to follow with natural numbers are:
x + y = y + x The commutative law of addition
x * y = y * x The commutative law of multiplication
(x + y) + z = x + (y + z) The associative law of addition
(x * y) * z = x * (y * z) The associative law of multiplication
(x + y) * z = x * y + y * z The distributive law
The Integers: It includes all the natural numbers, the zero number and all the negatives of the natural numbers. The example for this is : ..., -3, -2, -1, 0, 1, 2, 3, ....
Rational Numbers: The rational numbers can be defined as the ratios of integers. The most important condition is that denominator can never be a zero.
Real Numbers: It is basically a decimal expression whose digits or values may or may not terminate or repeat.




Let’s talk about scientific Notation: It is a way of writing numbers that accommodates or houses values too large or small to be conveniently written in standard decimal notation. Or in simple language a Scientific notation is used to express very large or very small numbers.

A number in scientific notation is written as the product of a number (integer or decimal) and a power of 10. This number is always 1 or more and less than 10. Scientific Notation is based on powers of the base number 10.
The number 33,000,000,000 in scientific notation is written as:
3.3 x 10¹¹
Here the first number 3.3 is called the coefficient. It must be greater than or equal to 1 and less than 10.
The second number is called as the base value. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. The number 11 is referred to as the exponent or power of ten.

Now the another section that is Logarithms:
The logarithm of a number is basically an exponent of a number or we can say that an exponent by which another fixed value, the base of the value, need to be raised to produce that number. In simple mathematical manner we can say that the logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. In mathematical view : b^y = x.

Now we are going to discuss about Logarithmic Functions in mathematical world. We can define logarithmic function as the inverse of the exponential function. In the above formula:
log_b(x) which is equivalent to x = b^y.
The logarithm to the base e (exponent) is written as l_n(x).

A brief introduction on Radicals:
Radical is known as root of an expression. A fraction is not a radical, but a fraction may contain a radical. It uses the sign of radical (√) also known as surds. Let's take an example to understand it better. A radical equation is an equation in which at least one variable expression is stuck inside a radical, usually a square root.
squares 2 = 4 it means 2 is the square root of four. 2³ = 8 this means 2 is the cube root of 8. if a, b are real numbers, n is a positive integer and if aⁿ = b then the n th root of b is a. then it can be written in this form : n root b = a
In n root b, n is the indexed and b is the radicand. The index gives the degree of the roots.

For example : Root x + 2 = 4
root x = 4
root x = squared 2
x = 2.
This is all about basic ninth grade Algebra, moving further we all are going to learn equation section of ninth grade algebra.

Introduction to IX Grade Algebra

Friends today we all are going to rewind our basic concepts of Algebra along with ninth standard Algebra problems. We all know about the basic concepts behind numbers and how can we combine them with the help of basic operations of addition, multiplication, subtraction and division. This mathematics area of study is known as Arithmetic. In ninth standard we are going to study the more advanced area of Algebra which is distinct from arithmetic. In this section of Algebra we are going to perform basic operations for example addition to specific numbers that involves entities called as variables. Variables are those entities which have no particular value or an unknown value. The variables in mathematics are usually denoted by the upper or lower case letters. Algebra is a fundamental branch of mathematics that mainly deals with the rules of operations and their relations and the constructions and concepts arising from them. If we talk in simple words then we will see that algebra is simply the art of replacing variables in place of numbers.

Now I am going to discuss it in deep. An algebraic expression is basically a collection of numbers and letters combined by the four (addition, multiplication, subtraction and division) basic arithmetic operations. Here are some of the examples of algebraic expressions to understand it better.

7y, 3a+y, 3a-4y, a/(a+y), a², (a+b)²
The numbers used in the algebraic expression are called constants. Additionally, if there is no variable in the algebraic expression then we called it an arithmetic expression. For example 4 + 3 / 8.

Variables in the algebraic expressions are basically used to describe general conditions or situations or we can say that it models the real problems in the form of algebraic expressions and they can be used to solve problems that in other manner would be much more difficult or complicated or even impossible. These applications are used when we all are going to solve word problems.

Now moving forward let’s discuss about one of the most important and frequently used terminology that is an equation. An equation is basically an allegation that two algebraic expressions are equal or similar. This statement can be represented in two different ways or we can say that it can have two distinct meanings:

The first statement says that the equation is true for all values of the variables. In such kind of situations the equation is called an identity. An example of an identity to understand the above statement is
x + y = y + x
for all numbers x and y. This statement or example is called the commutative law of addition. Another well known identity is the first binomial formula which can be stated as :
 (a+b)² = a= + 2ab + b²
The second statement says that the equation is true for some of the values of the variables. Here we need to find out the values of the variables for which the equation is true. This terminology is called as solving the equation. Let’s take an example to understand it better. The equation to solve is: 3y + 1 = 4
in this equation we know that
y = 1
is the solution. The above equation is an example of a linear equation. Let’s take a bit more complicated example of a quadratic equation is y² - y - 2 = 0. Now after solving this we found that this equation has the two solutions
  y = -1    or   y = 2.
Now in Ninth standard we are also going to understand the basic concept behind evaluating an algebraic expression. To evaluate an algebraic expression can be stated as, substituting specific values for its variables. Lets’ take a very simple example to understand it:
Expression = 2y + 1
Evaluating expression, for the value of variable y = 3
give expression = 2 * 3 + 1= 7
We can say that value of expression at y = 3 is 7.
Let’s take an another way to solve which is a bit more complicated:
Expression at y = 2x + 1, where x is another variable
Expression = 2 * (2x + 1) + 1 = 4x + 2 + 1 = 4x + 3.

The next condition is if there are equivalent expressions:

Two expressions are equivalent if their values are equal for all possible estimations of the two expressions. In a simple mathematical manner we can say that listing expressions with an equality sign between them always gives an identity.

Now the next topic of the Algebra section is Number system. This section tells about the different types of numbers. Here we are going to talk about the four types of numbers that are Natural numbers, the Integers, rational numbers and real numbers.
Natural Numbers: 1, 2, 3, 4, 5, ..... are the natural numbers. We can add or multiply the two given natural numbers and obtain another natural number. Some of the mathematical laws to follow with natural numbers are:
x + y = y + x The commutative law of addition
x * y = y * x The commutative law of multiplication
(x + y) + z = x + (y + z) The associative law of addition
(x * y) * z = x * (y * z) The associative law of multiplication
(x + y) * z = x * y + y * z The distributive law
The Integers: It includes all the natural numbers, the zero number and all the negatives of the natural numbers. The example for this is : ..., -3, -2, -1, 0, 1, 2, 3, ....
Rational Numbers: The rational numbers can be defined as the ratios of integers. The most important condition is that denominator can never be a zero.
Real Numbers: It is basically a decimal expression whose digits or values may or may not terminate or repeat.




Let’s talk about scientific Notation: It is a way of writing numbers that accommodates or houses values too large or small to be conveniently written in standard decimal notation. Or in simple language a Scientific notation is used to express very large or very small numbers.

A number in scientific notation is written as the product of a number (integer or decimal) and a power of 10. This number is always 1 or more and less than 10. Scientific Notation is based on powers of the base number 10.
The number 33,000,000,000 in scientific notation is written as:
3.3 x 10¹¹
Here the first number 3.3 is called the coefficient. It must be greater than or equal to 1 and less than 10.
The second number is called as the base value. It must always be 10 in scientific notation. The base number 10 is always written in exponent form. The number 11 is referred to as the exponent or power of ten.

Now the another section that is Logarithms:
The logarithm of a number is basically an exponent of a number or we can say that an exponent by which another fixed value, the base of the value, need to be raised to produce that number. In simple mathematical manner we can say that the logarithm of a number x with respect to base b is the exponent to which b has to be raised to yield x. In mathematical view : b^y = x.

Now we are going to discuss about Logarithmic Functions in mathematical world. We can define logarithmic function as the inverse of the exponential function. In the above formula:
log_b(x) which is equivalent to x = b^y.
The logarithm to the base e (exponent) is written as l_n(x).

A brief introduction on Radicals:
Radical is known as root of an expression. A fraction is not a radical, but a fraction may contain a radical. It uses the sign of radical (√) also known as surds. Let's take an example to understand it better. A radical equation is an equation in which at least one variable expression is stuck inside a radical, usually a square root.
squares 2 = 4 it means 2 is the square root of four. 2³ = 8 this means 2 is the cube root of 8. if a, b are real numbers, n is a positive integer and if aⁿ = b then the n th root of b is a. then it can be written in this form : n root b = a
In n root b, n is the indexed and b is the radicand. The index gives the degree of the roots.

For example : Root x + 2 = 4
root x = 4
root x = squared 2
x = 2.
This is all about basic ninth grade Algebra, moving further we all are going to learn equation section of ninth grade algebra.

Ninth Standard math introduction

Friends, today we all are going to discuss about the overall syllabus of your grade IX mathematics and also going to talk about the most suitable way to solve such type of problems. Sadly, most students start learning mathematics by assuming it as a bundle of recipes to handle certain problems. Eventually mathematical world is a web of facts, concepts, and logical reasoning and, a more efficient approach to understand it is to focus on the principles and connections rather than on isolated facts.

The first approach to understand this important concept are building mathematics in this what students need to do is to reduce their problem to one that he/she have solved before. This principle is used all around in mathematics. The main sayings of this principle is : in order to understand a piece of mathematics student have to understand what preceded it. Moving forward what student need to do is to introduce concepts in a simple context and then generalize them in such a manner that facts and rules that are true in the simple context always remain true in the more general context. Before student go and attempt the solution of a given problem or starts studying a subject, it is worthwhile to think absolutely about what he/she expect their solutions to look like, or what they expect to learn. In general there are two outputs or we can say that two possibilities that are students expectations are met or they are not. In the second situation what students generally do is they make a mistake and now that they are alert to the fact that they can recover from it.

Now I am going to give you an introduction about your IX grade mathematics syllabus. The ninth grade syllabus mostly includes following topics in most generalized manner. The topics to cover are as follows :

1. Algebra
2. Geometry
3. Statistics
4. Probability
5. Pre calculus
6. Calculus
7. Trigonometry

Firstly discuss about the Algebra topic of ninth standard, This topic typically treated in ninth standard and most of the topics are considered as the part of Algebra 1. It includes exponents, radicals, properties of numbers, patterns, number system, evaluation of expressions involving signed numbers, exponents and roots, properties of the real numbers, absolute value and equations and inequalities involving absolute value, scientific notation, unit conversions, solution of equations in one unknown and solution of simultaneous equations, the algebra of polynomials and rational expressions, word problems requiring algebra for their solution, Pythagorean theorem, algebraic proofs, functions and functional notation, solution of quadratic equations via factoring and completing the square, direct and inverse variation, and exponential growth. Some of the few other topics are Gauss-Jordan elimination and kinds of matrix formation that includes Matrix/vector addition and Matrix/vector multiplication.

Now let's have a brief introduction about how to solve a word problem. What student need to do is to look for the statements in the problems that describe equal quantities. Then use of algebraic phrases and equal signs to write equations that make the same statements of equality.

To solve a difficult problem what student need to do is to solve a simpler problem first and then apply whatever formulas or methodologies that student learn to solve the more difficult problem. If student is going to opt this technique then he/she might have to iterate this technique and try to build whole hierarchy of problems to solve a truly difficult problem.

Moving further let's talk about the next topic in the ninth grade mathematics, that is Geometry. Geometry is an important area of mathematics which deals with the shape, size, relative position of figures, and the properties of space. It is all about shapes and their properties. Geometry is of two types : Plane geometry and solid geometry. Plane geometry deals about the shapes on a flat surface like lines, circles and triangles, shapes that can be drawn on a piece of paper whereas Solid geometry is all about three dimensional objects like cubes, prisms and pyramids. In ninth standard Geometry, we all are going to learn the following sub topics that are : Euclidean theories and algorithms, then we study about properties of geometry and a whole concepts of coordinates and transformations. Another important topic which is used in both Algebra and in Geometry that is Pythagorean theorem which helps in calculating the third side in a right angle triangle. Then we are going to understand the basic concept behind the coordinate system and look towards measurement of Geometry like triangle, square etc. In addition to this we will talk about different Geometric concepts and isoceles triangle theorems and rectangular coordinate system. Another important section comes under Geometry is Rate, distance, time, angle measurement and arc length. Inequality theorem along with angles of triangles and polygons along with unit of measures etc. are some of the important topics of Geometry.
Now the next topic of ninth standard mathematics are Statistics and Probability which are interrelated with each other, so I am going to discuss both the topics together. In lots of statistical analysis, results depend on the probability calculation like in many of the census results depend on probability and to represent probability we need statistics help. Probability and Statistics together play an important role in finding out measures of central value, measures of spread, and helps in comparing of two data. These two branches are basically treated as the tools of data analysis. Statistics is the practice or science of collecting and analyzing numerical data in large quantities. It is basically a study of the collection, organization, analysis and interpretation of the data. Probability is a way of telling or expressing knowledge that an event will occur or has occurred. The probability of an event occurring given that another event has already occurred is called a conditional probability. So in this section we deal with Mean, median, mode, range, standard deviation and variance in the numeric sequences. We are going to find the combinations and permutations among the series of numbers along with this we need to represent the data with the help of Data representation methods. In probability section I am going to discuss about types of events, conditional probability etc. Statistical analysis and methods to make inferences and organizing and displaying data along with making predictions from data are the another important topic of ninth grade mathematics.
Now the other two topics related with each other are Pre-Calculus and Calculus. Calculus is basically a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Basically it is the study of 'Rates of Change'. There are two main branches of calculus: Differential Calculus and Integral Calculus. Differential calculus determines the rate of change of a quantity and integral calculus finds the quantity where the rate of change is known. In ninth standard Pre-Calculus section I am going to discuss about the following topics that covers topics like describing functions, representing functions along with Arithmetic and geometric series. The calculus section of mathematics branch includes various mathematical models with their applications to solve problems using multiple approaches along with this it also deals with techniques used to study patterns and analyze data and problems related to personal income and credit. It also comes with problems related to polar and rectangular coordinates along with trigonometric form of complex numbers and mathematical induction. Conic section and fundamental theorem of algebra are also the part of Calculus problem.
The last and final sub topic of ninth grade mathematics is Trigonometry. It is one of the important branch of maths and it also plays an important role in the physics branch. Trigonometry is the branch of mathematics that deals with triangles, circles, oscillations and waves. In mathematical term it is the study of triangles and the relationships between their sides and the angles between these sides. Trigonometric functions are the functions of an angle of a triangle, that are used to relate the angles of a triangle to the lengths of the sides of a triangle. The sub topic of ninth standard trigonometry are Degree and radian measures along with various basic trigonometric functions which I already stated above. Special right triangles means triangles placed in different form or angles and trigonometric values of standard angles are the next important topic of mathematics. Justification of Pythagorean identities are also an important section along with different trigonometric equations is also included in the syllabus.
Whenever student is going through several problems then he should go for work on the easy one first. As this might teach the students something that will surely help them to solve the more difficult problems in a faster and in a better manner. Whenever a student is done with their mathematic question, he or she needs to check the answers as there may be some mistakes in the problem or solution obtained is not an appropriate one. So moving further in next class we again start from the first topic of ninth grade mathematics and try to understand the basic concept behind the Algebra.

Logarithmic Rules and formulas in IX Grade

Mathematics is one of the engrossing subject, still students try to move away from this wonderful subject.  But, my dear friends now, onwards you don’t need to move away from this subject as I am here to help my kids, so that they can easily grasp the concepts of this subject. Now, I will teach you all the math topics. Grade IX is the most important subject of student’s career as it decides where they have to go after this. In this article we will talk about the very crucial topic of algebra taught to students of class IX. Grade IX is the starting class of higher studies and from this class subject become tough and complex. If one has intention then he/she can easily reach to its goal with little efforts. So set your target and start making all possible efforts to reach it. Ok, now, move to the topic that we have to study today, the Logarithmic rules and formulas. Logarithmic is the new topic introduced in Grade IX.

What do you understand by this term what did it mean? No idea, ok , I will tell what is this, Logarithms is an exponent to which constant is raised to obtain a given number. In other words, we can also define it as:
The logarithm of a number represents the quantity as a power to which a fixed number has to be raised to produce a given number. The fixed number is known as base. For example, the logarithm 10000 to base is 4, because 10000 is 10 to the raised power 4 that means:
10000 = 10⁴ = 10 x 10 x 10 x 10.
One more general from we can define it as if x = c^y  then y is the logarithm of x to base b, and is written log_c(x), so log₁₀(10000) = 4.
x = c^y is the same as y = log_bx

In simple words logarithm is just an exponent. Kids don’t get confused, let’s specify it, the logarithm of a number x to base b is just is just the exponent you put onto b to make the result equal x.
Let’s have an example to understand this concept, as you know 6² = 36, is in exponent form (the power is 2) and this is the logarithm of 25 to base 5. In above paragraph I explained you how to write log function, log₆(36) = 2.
Let’s see an example to understand the basic concept of logarithm.
Log₃81 = ?  is the same as 3? = 81,
In the second equation if we calculate the value of unknown than we can easily the value of equation one.
blog b_x = x,

We can read it as the logarithm of x in the base b is the exponent you put on b to determine the value of x as a result.
3x = 81,
x = 81/3,
x = 27,
Now, we can easily put the value of known in the equation 1,
Log₃81 = 27.
In this way we can easily determine the value of unknown in log.

In mathematics we deal with the two types of logarithms: common logarithm and the natural logarithm. The general definition of log is given as: log_ax = N means that a^N = x.
The common logarithm is given as: log_x = log₁₀x. in common log a logarithm is written without a base.

In natural logarithm the logarithm function is written ln and it is given as: ln x = log_ex where, e’s value is approximately 2.718.
For the above statements let x, y, a and b all are positive and a ≠ 1 and b ≠1.

Each language has its well – defined set of rules and if you will not follow the rules properly then you will get wrong result. In the same way mathematics has its set of rules and its topic contains a proper method and rules that you have to follow to get the correct answer. As, in this article we are talking about logarithm so now talk about, Logarithmic rules.

The first rule is known as the inverse property rule that depicts:
log_aa^x = x   and   a(log_ax) = x

Is the product rule that says that: log_a(xy)= log_ax + log_ay

Now, move towards the example so that you can understand what I want to explain you in above theory. Like: expand log₃(2x),
Whenever any problem related to expansion arises it means that you have one log expression with lots of stuff inside it and you have to use the log rules and functions to get the complicated answer. In the above question we have 2x inside the log problem. In this question 2x is in multiplication form so apply the multiplication or product rule to get the answer. When we apply product rule we get:
log₃(2x) = log₃(2) + log₃(x).  And this is the answer of the problem.


The third one is defined as the Quotient rule and according to it:

.

For illustration of the above property let’s take an example and solve it with this you came to know how to use division or quotient rule.

Simplify: log₄(16/x).

In this you can easily find that we have to apply division rule as the logarithmic function contains (16/x) in division form.
According to division rule we get:
log₄(16/x) =  log₄(16) - log₄(x),
The first term on the right hand side can be simplified to exact form or value, by applying the basic rule of the log that we discussed in the definition of the log.
log₄(16) = 2, so, the final answer is:
log₄(16/x) = 2 - log₄(x).
After you get the answer don’t forget to cross check it and see that you properly expand all the terms or not.

The fourth rule is known as the power rule that shows:
log_a(xp) = p log_ax.
To understand this property properly let’s take an example and solve it. log₅(x³)
The exponent inside the log can be taken out as a multiplier
log₅(x³) = 3.log₅(x) = 3log₅(x)
This is the simple property and according to this rule we take the power in the front side of log as a multiplier. This is simple one but most commonly used property of log used in almost each and every problem.


The last one is defined as the change of base formula:

Example for this: log₃[4(x - 5)² / x⁴ (x -1)³],
In this we will solve the question using the above property. In the first step we proceed as:
log₃[4(x - 5)² / x4 (x -1)₃] = log₃[4(x- 5)₂] – _log3[x4(x -1)³],
                                         =[ log₃(4) + log₃[(x -5)²)] - log₃(x⁴) + log₃[(x - 1)³)]
                                         = log₃(4) + log₃[[(x - 5)²] - log₃(x⁴) – log₃ [(x - 1)³]
                                      = log₃(4) + 2 log₃[ (x -5)] - 4log₃ (x) - 3log₃[(x - 1).
Thus, this is the final answer of the above problem.

While solving logarithmic problem you all must keep few things in mind so that you can solve the question properly without any problems without mistakes.
log_a (x + y)   ≠ log_ax + log_ay,
log_a(x - y)   ≠ log_ax – log_ay.
Now, let’s see one more example that students can understand the different type of problem that occurs in logarithms and as you know example is the best way to learn and understand different concepts rule or functions.  So here is one more example for you and see the step involved while solving the problem and with this you can easily practice lots of questions,
log₂ (8x⁴ / 5)
Solution:
Step 1: 5 is divided into the 8x⁴ , so first split the numerator and denominator by using subtraction, on doing this we get:
log₂(8x⁴ / 5) = log₂ (8x+) - log₂(5),
Step 2: in the next step don’t take the exponents out; it is only on the x, not on the 8 and we can take only those exponents that are present on every term inside the log. So in this split the factors using the addition,
log₂ (8x⁴ ) - log₂(5)= log₂(8) + log₂(x⁴) - log₂(5),
Step 3: the x has an exponent so take it in front of log as a multiplier,
log₂(8) + 4log₂(x) - log₂(5),
Since the power of 2 is 8, I can simplify the first log to an exact value:
log₂(8 ) + 4log₂(x) - log₂(5),
3 + 4 log₂ (x ) - log₂(5)
Now, each log contains single value so the expression is simplified and the answer of the problem is, 3 + 4 log₂(x ) - log₂(5).
This is all about logarithmic rule and logarithm formula and for detail switch to online help.