Why Algebra is So Important? Do You Afraid Algebra? Learn Proven Techniques to Understand Algebra to Score Maximum Marks in Examinations?
- What's Algebra
- Types of Algebra
- Important Topics of Algebra
- Applications of Algebra
- Definition of Useful Basic Terminologies
- Significance of Algebra
- Branches of Algebra
- Techniques of Mastering Algebra
- Applications of Algebra in Real Life
- How to Score Maximum Marks in the Algebra
- Frequently Asked Questions on Algebra
- Conclusions
1. What's Algebra?
Algebra is the branch of mathematics that helps in the representation of problems or situations in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, subtraction, multiplication, and division to form a meaningful mathematical expression. All the branches of mathematics such as trigonometry, calculus, and coordinate geometry, involve the use of algebra.
Examples:
We can express the algebraic equations with the help of variables like
1) 4x-6=12
2) x+y =10,...(1)
x-y=4....,(2)
2. Branches of Algebra and Important Topics
There are five different branches of algebra: elementary,
advanced,
abstract,
linear, and
cumulative.
Let us know about these branches in depth.
Elementary Algebra
This branch of mathematics is concerned with the fundamental properties of numbers, variables, constants, and their relationships.
Equations, formation, manipulation, expression evaluation, equalities, inequalities, equation solutions (both algebraic and linear), and other topics are covered in elementary algebra.
It also enables the standard formulation of arithmetic principles such as x + y = y + x, and it is the first step in demonstrating the systematic study of all of the properties for a system of real numbers.
Advanced Algebra
This is Algebra's intermediate level. In comparison to pre or elementary algebra, this algebra has a lot of equations to answer. Advanced algebra will assist you in understanding other aspects of algebra, such as:
Matrices
Series
Trigonometry
Equalities
Conic Sections
Polynomial Situations
Inequalities
Sequences
Probability
Graphic Representation
Rational Expression
Linear Algebra
Linear algebra is an area of mathematics that has applications in both applied and pure mathematics. It is concerned with the linear mappings of vector spaces. The study of planes and lines is also part of it. It is the study of linear equation sets that have transformational features. It is universally used in mathematics. It deals with linear equations and their representation in vector spaces and matrices for linear functions.
The following are some of the most essential subjects studied in linear algebra:
Linear Equations
Matrices
Matrix Decomposition
Vector Spaces
Relations
Relations and Computation
Cumulative Algebra
It is a branch of algebra that investigates commutative rings and their respective ideals. Commutative algebra is required for algebraic number theory and algebraic geometry. Rings of algebraic integers, polynomial rings, and so on are examples of this branch. Commutative algebra is used in many other fields of mathematics, including differential topology, invariant theory, order theory, and general topology. In modern pure mathematics, it has played a significant role.
Abstract Algebra
Abstract algebra is one of the divisions of algebra that seeks to discover truths about algebraic systems that are independent of the nature of particular operations. In some circumstances, these operations have particular features. As a result, we can draw certain conclusions about the ramifications of such features. Fields, groups, modules, rings, lattices, vector spaces, and other algebraic structures are studied in abstract algebra.
The abstract algebra concepts are as follows:
Binary Operations
The binary operations are created when the concept of addition is imagined. Without a set, the concept of all binary operations will be useless.
Inverse Elements
The term "inverse elements" refers to a concept that involves a negative integer. In addition, we write “-a” as the inverse of “a,” and for multiplication, we write “a-1′′ as the inverse form.
Sets
A set is defined as a collection of things whose existence is determined by a given property. For example, a collection of all 2 x 2 matrices, a collection of two-dimensional vectors in the plane, and many types of finite groups.
Associative
When integers are added together, there is a property known as associativity, which states that grouping the numbers does not impact the sum. Consider the following equation: (9 + 6) + 3 = 9 + (6 + 3).
Identity Element
For a certain operation, the numbers 0 and 1 are viewed as an identity element. For the addition operation, 0 is the identity element, but for the multiplication operation, 1 is the identity element.
3. Applications of Algebra
Just as multiplying two by twelve is faster than counting to 24 or adding 2 twelve times, algebra helps us solve problems more quickly and easily than we could otherwise. Algebra also opens up whole new areas of life problems, such as graphing curves that cannot be solved with only foundational math skills.
What are the application of algebra in different fields?
Fields of application of algebra
The Common examples & applications of algebra use in daily life are the following
- Catch the ball game played by 4-5-year-old kids.
- Making a schedule of activities.
- Preparing the food
- Doubling or halving the recipe.
- A kid developing spatial intelligence.
- Finding the tax liability.
- Astrological calculations.
- Technological developments.
- Budgeting.
Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems.
Some examples are given below.
If the square of the number x is multiplied by 4, and then 15 is added to that product, the result can be represented by 4x2 + 15
If Shyam’s present salary s is increased by 15 percent, then his new salary is 1.15s.
If y gallons of petrol are to be distributed among 5 people so that one particular person gets 1 gallon and the rest of the petrol is divided equally among the remaining 4, then each of those 4 people will get
(y − 1) ÷ 4
4. Algebra Terms and Definition
The following are the basic terminologies, you must know like : constants, variables, coefficients, terms, expressions, and equations etc.
Constants
A value or number that never changes in an equation — it’s constantly the same.
For example,
the number 5 is a constant because it is what it is; 5 is never 23 or some other number — 5 is always 5. A variable can be a constant if it is assigned a definite value. Usually, a variable representing a constant is one of the first letters in the alphabet.
For example, in the equation ax + bx + c + 0, a, b, and c are constants and the x is the variable. The value of x depends on what a, b, and c are assigned to be.A fixed quantity that does not change. For example: 3, –6, Ï€, 1/2
Variables
A variable is a symbol that we assign to an unknown value. It is usually represented by letters such as x, y, or t. For example, we might say that l stands for the length of a rectangle and w stands for the width of the rectangle.
We use variables when we need to indicate how objects are related even though we may not know the exact values of the objects.
For example,
Coefficients
The coefficient of a variable is the number that is placed in front of a variable.
For example,
Exponent.
A small number written slightly above and to the right of a variable or number. It’s used to show repeated multiplication. An exponent is also called the power of the value.
Constant.
A value or number that never changes in an equation — it’s constantly the same.
For example, the number 5 is a constant because it is what it is; 5 is never 23 or some other number — 5 is always 5. A variable can be a constant if it is assigned a definite value. Usually, a variable representing a constant is one of the first letters in the alphabet.
For example,
Equation. A grouping of terms or values that use a sign to show an equal relationship — that two quantities are equal. By using an equation, tough problems can be reduced to easier problems and simpler answers.
·Exponent. A small number written slightly above and to the right of a variable or number. It’s used to show repeated multiplication. An exponent is also called the power of the value.
Expression. Any combination of values and operations that can be used to show how things belong together and compare to one another.
Factor. To change two or more terms to just one term so that you can perform other processes. For example,
ab + ac +ad = a(b + c + d)
Operation. An action performed upon one or two numbers to produce a resulting number. Operations are addition, subtraction, multiplication, division, square root, and so on.
For example, in the equation 3 + 4 = 7, the operation called for is addition.
Simplify. To combine all that can be combined, reduce the number of terms, and put an expression in an easily understandable form.
One example is to simplify the fraction 2/4 to 1/2.
Solve. To find the answer. In algebra, it means to figure out what the variable stands for.
For example, if you solve for x in the equation 4x = 20, you need to determine what number the variable x represents. So, divide 20 by 4, and you've solved the equation: x = 5!
Terms
A term can be any of the following:
a constant: e.g. 3, 10, π, 1/2
the product of a number (coefficient) and a variable : e.g. –3x, 11y, 2/3 a
the product of two or more variables: e.g. x2, xy, 2y2, 7xy
Like terms are terms that differ only in their numerical coefficients.
For example: 3a, 22a, 2/3 a are like terms.
Expressions
An expression is made up of one or more terms.
For Example:
3w + 4xy + 5
Equations
An equation consists of two expressions separated by an equal sign. The expression on one side of the equal sign has the same value as the expression on the other side.
For Example:
8 + 6 = 7 × 2
l = 3 × w
3w + 4xy + 5 = 2w + 3
Quadratic Equations
A Quadratic Equation is an equation of the form:
ax2 + bx + c = 0, where a, b and c are numbers and a ≠ 0
For Example:
Algebraic Fraction
An algebraic fraction is a fraction that contains an algebraic expression in its numerator and/or denominator. For example:
42x−3, 3x−5x+3
5. How to Understand Algebraic Variables
In algebra, variables are placeholder letters (capitalized and lowercase) that represent the unknown, or what you’re solving for.
Examples
6. Importance of Algebra
These reasons go far beyond the walls of the classroom as math is used more often in everyday life than most people realize.
1. Applications of Algebra in Other Subjects
The first and most important reason algebra is important is that it can be applied to other subjects.
These include:
Chemistry
Biology
Social sciences
Engineering
Physics
Food science
Psychology
Other math classes like geometry, calculus, and trigonometry also use algebra as a base, making it an essential skill to master. For this reason, you’ll find that grades can improve across the board by improving math skills.
2. Use of Formulas to solve problems
Without algebra, we wouldn’t have an easy way to figure out the area of a shape. Algebra allows you to take any formula, plug in numbers, and solve an equation. This is also a huge part of why algebra is used in so many other subjects.
For example, if you need to figure out how much paint is needed to cover a triangular-shaped wall, you need algebra to figure it out. Rather than guessing and buying way too little paint, you’ll know exactly what you need.
3. Simplify Complex Problems
Imagine if we didn’t have multiplication. Multiplication is a shortcut that allows you to simply figure out 5 x 10. If you couldn’t multiply, you’d have to add all those numbers together, or count out a grid with 50 points on it.
In the same way, algebra simplifies problems so that they’re more easily solved. As we mentioned before, this is usually done with formulas.
4. Solve Problems Quickly
As you can already imagine, being able to solve problems with a simple algebraic formula is far faster than other methods. This not only allows homework to go faster but everyday situations.
If you know the formula, you know all you have to do is multiply the height by the length of the base and then divide by two.
5. Better Understand Numbers
Algebra teaches you to follow a logical path to solve a problem. This, in turn, allows you to have a better understanding of how numbers function and work together in an equation. By having a better understanding of numbers, you’ll be better able to do any type of math.
6. Intricate Part of History and Culture
Math has been around since humans could count and started bartering. Since then, it’s evolved right alongside us and is an intricate part of our history and culture.
Without math, humans might not be at the top of the food chain. We certainly wouldn’t have come as far as we have. So, by learning algebra, you can gain a better appreciation for our human history.
7. Countless Career Options
There are countless careers that require an understanding of math and algebra in particular.
These include:
Computer programmers
Statisticians
Teachers
Doctors and veterinarians
Scientists
Mathematicians
Numerous other career paths will require you to take algebra and other advanced math classes in college to get your degree. For that reason alone, it’s vital to understand algebra at an early age.
8. Improve Technology
Speaking of careers, if you want to do anything to improve technology, you need algebra. This and other similar types of maths will give you the base necessary to help make technological advances.
For example, it was the use of maths that allowed us to put astronauts on the moon. Without it, the astronauts wouldn’t have been able to calculate the angle at which to leave the earth’s atmosphere.
Today, algebra is used in nearly every aspect of technology, which includes how smartphones and computers operate.
9. Make Wiser Decisions, Save Money
This is a great example of how algebra can help you in your everyday life. By applying algebra to your budget and spending habits, you can make wiser decisions with your spending.
By going through your bills and exploring other options, you can find out which company is actually giving you the best deal. You can use this on everything from insurance to your cell phone bill to save money.
10. Helps build your confidence
The final reason why learning algebra is important is that it challenges you. It will teach you how to face struggles and to overcome them.
When you overcome a challenge, it helps build your confidence so you’re ready to face the next one. There are many challenges in life you’ll face that are more difficult than algebra, but it’s a good place to start.
Now most important for examination point of view:
The Winning Tips and Tricks of Mastering Algebra
Below are 14 study tips and common pitfalls to avoid to help you succeed in algebra.
1. Know Your Arithmetic
To learn algebra, you have to, have to, HAVE TO know your basic arithmetic. Yes, that includes addition, subtraction, multiplication, and division.
2. Remember PEMDAS or BODMAS : Both are Same
It's order of mathematics operation. You learn it. Take a test. Then delete it from your memory thinking it won’t show up again.
This is where PEMDAS comes in handy.
Parentheses, Exponents, Multiplication, Division, Addition, Subtraction
The process of PEMDAS is given in the image below. Follow the steps
Here’s one thing to watch out for when you use PEMDAS that many people miss:
After you take care of your parentheses and exponents, you do multiplication and division in order from left to right. Then, when you move on to addition and subtraction, you do those in order from left to right as well.
Here’s an example of a problem that may seem difficult at first but becomes completely manageable using PEMDAS:
5 + (6 ÷ 2 x 5)2 – 2 + 3
Using PEMDAS, we start out with the expression inside the parentheses. There are no exponents inside the parentheses, so we go next to multiplication and division, working from left to right. This means you will do the division first:
5 + (3 x 5)2 – 2 + 3
Now multiply inside the parentheses:
5 + (15)2 – 2 + 3
After you’re done with the parentheses, the next step is to handle the exponent in (15)2:
5 + 225 – 2 +3
At this point, there isn’t any multiplication or division left, so we can skip right to the add and subtract steps from left to right:
230 – 2 + 3
228 + 3 = 231
3. Positively or Negative Numbers: Be Comfortable
Negative numbers are just like the numbers you know and love but, well, negative. Don’t let that tiny minus sign in front throw you.
Be sure to get comfortable with the basic operations of negative numbers. That is, adding, subtracting, multiplying, and dividing two negative numbers AND a positive and a negative number.
This will be extremely helpful down the line!
+X+=+
+X- =-
-X+=-
-X-=+
Here are a few examples and rules to remember:
a. When numbers have the same sign, add them and keep the original sign.
-3 – 5 = -3 + (-5) = -8
b. When numbers have different signs, add them and keep the sign of the “larger” number, or the number with the greatest distance from zero.
-10 + 7 = -3
c. A negative divided by a negative is a positive.
-4 ÷ -2 = 2
d. A negative multiplied by a positive is a negative.
-15 x 3 = -45
4. Let's Others Judge Your Work
A lot of students try to rush through algebra problems. But taking the time to show all your steps not only helps you keep organized and avoid small errors, but you might even get partial credit for a wrong answer if your instructor can see that you were on the right track.
We know: Writing down all your steps takes a little longer. But one small mistake in an algebra problem can lead to a big headache in the end.
5. Don’t Afraid with Letters
For most of your early math career, you learned that math is all about numbers. But then algebra threw x’s, y’s, and sometimes even z’s at you. Be assured they’re not as bad as they look! These letters are called variables and they really are numbers too.
Yes, it can be confusing at first, but it does get better! Mathematicians use letters as variables to stand for things they want to find.
Think of a simple problem that asks, “What is two times three?” The “what” part is a variable. It’s not necessary, but we could write the same problem as:
x = 2 x 3
where is the “what” we want to solve for.
If a problem has more than one variable, that just means there are more things to find! In these cases, don’t jump the gun. Find one variable at a time.
6. Make Formulas Your Friends
Think of formulas like a team of helpful sidekicks, especially when it comes to word problems. Depending on your class, there are a number of formulas you will need to memorize and be able to use.
It’s important to know that just memorizing the formula doesn’t necessarily mean you’ll know what to do with it! Be sure to know what each variable in the formula stands for so that you can decipher what number is assigned to which variable.
Some of the common formulas you will see are:
7. Answer the Easy Questions First
Don’t circle your answer and call it a day until you double check that you have what the problem is asking for. You may be wondering why this is even a tip. But this is a common error in algebra!
For example, some problems may ask you to find the dimensions of a box and you’ve only found the length. Just because you have at the bottom of your paper doesn’t mean you have the answer.
8. Work Out How to Practice on Problems
One last tip for studying is to work out so many problems your hand gets tired. This goes for all math subjects but especially applies to algebra. The way to perfect your problem-solving ability is to practice, practice, practice!
9. Don't Worry, Learn from Your Mistakes
Don’t be afraid to make mistakes! They’re part of the learning process. Even seasoned mathematicians make them. When you do make a mistake, the key is learning to recognize there is a problem, identify your mistake, and then correct it. That’s where real learning occurs.
10. Don’t Divide By Zero
You can’t do it. You will never be able to do it. So don’t do it. It’s one of the first rules in the Holy Book of Mathematics: “Thou shalt not divide thee by zero.”
11. Always Follow the rules of Parentheses
Even though they might not seem like it, parentheses are important.
Very important.
They tell you a lot about the problem such as what to do first or what to group together.
12.Do not Distribution the Exponent
Here’s one of the most common mistakes in algebra:
Problem:
Expand (x + y)2
Common student solution:
x2 + y2 => WRONG!
If there is an exponent on the outside of parentheses and addition or subtraction on the inside of the parentheses, YOU CANNOT DISTRIBUTE THE EXPONENT.
Don’t do it.
13. Only Cancel Factors
You can only cancel a factor of the numerator with a factor of the denominator if they are exactly the same.
A very important part of this sentence is the word factor. To cancel terms, they must be factors. A term is a factor if it is being multiplied by everything in the expression.
14. Don’t Forget to Distribute Parentheses
One more quick thing before you leave! If there is an x in front of parentheses that have addition or subtraction on the inside, that x needs to be multiplied to EACH term inside the parentheses. If there is a negative sign in front, that needs to be distributed the same way!
7. Real life application of algebra
Here are some simple examples that demonstrate the relevance of algebra in the real world.
Cooking or doubling or halving the recipe:
The kitchen is one of the places where Algebra is used in an interesting way. Especially, when you are making items like cake, vegetable soups, etc, where the correct combination of the ingredients is very crucial, Algebra comes to your rescue and helps you find the correct quantity of ingredients to make the food sufficient for different sizes of servings.
Variables are the quantities of ingredients and the number of servings. By using Algebraic intelligence, the cook ensures that not only is the food available enough but also that it tastes great. Cooking is so Algebraic!
Doing Interiors and Landscape designing:
Algebra is used not only in interior planning or in landscape design, but throughout the whole Architecture program. The designers use data such as measurements of the space and the size of the items to locate the correct elevation or to determine the collection of items to be used in space.
Computer Programming:
While coding, the programmer is making use of datasets, strings, and variables. They assign conditions in the use of these strings and establish relationships, which ultimately result in an action on the front end. This activity of coding, assigning values, and pre-defining the functions is actually you applying the knowledge of Algebra.
Business & Finance Management:
Business is as dependent on algebra as any other field. In order to calculate profits and losses, business owners use algebraic operations. A business person will use algebra to determine whether a piece of equipment does not lose its worth if it is in stock.
In the finance industry, exchange rates and interest rates are often represented algebraically; therefore, to carry out finances accurately, it is necessary to be familiar with algebraic operations.
Sports:
Sports aren't forgotten when it comes to using algebra. Almost all athletes unintentionally use algebra. Cricketers can smash sixes because they can calculate the force needed to hit the ball, and basketball players can calculate the trajectory needed to get a point.
Footballers calculate the force and distance required to score a goal in a similar way, and sprinters estimate the speed required to traverse the distance to reach the endpoint; thus, algebra is involved in every sport in some way.
Finding the tax liability:
People work to earn money as they grow older. They might need to figure out how much money they owe in taxes. Calculations such as determining the tax rebate cap and proportioning profits to determine how the income should be shared among various investment possibilities are part of the process.
Astrological calculations: Astrologers use orbital motions to forecast various occurrences. They try to figure out what the relationship is between the planet's rotation speed and its position after a set amount of time, and so on. All of these calculations are Algebra.
8. How to Score Good Marks in Algebr?
A few tips to follow on how to score good marks in maths are as follows-
1. Read your curriculum thoroughly before beginning your exam preparations. Based on their importance, make a note of all the key ideas and concepts.
2. Analyzing the syllabus well as how to score good marks in maths
3. Ask your teacher for clarification if there is anything in the syllabus that you need help understanding .
4. Use sample papers and follow your professor’s advice. It will assist you in comprehending how to finish a specific job or project that you are finding challenging to attempt.
5. Mark all the essential elements that have been listed in the syllabus before you start studying a particular chapter. You will be able to save time this way and use it to focus on the topics that are important from the standpoint of the exam.
6. Manage Your Time Judiciously
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