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 : by = 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:
logb(x) which is equivalent to x = by.
The logarithm to the base e (exponent) is written as ln(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.
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 : by = 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:
logb(x) which is equivalent to x = by.
The logarithm to the base e (exponent) is written as ln(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.
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