1977 Learners
People often say that every student should know algebra by the time they finish high school. But what really is algebra? Is it truly that important? And why do so many people find it challenging to learn? First, algebra isn’t just “doing arithmetic with letters”. In reality, arithmetic and algebra are two distinct approaches to examining and solving numerical problems.
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Algebra refers to a mathematics discipline that teaches students how to work with variables, constants, and operations to model or solve a problem.
Algebra provides a systematic process of mathematically expressing the relationships and patterns. In algebra, students learn to represent unknowns and understand functional relationships and to analyze problems involving numbers and symbols. Algebra develops students’ ability to think abstractly about math, which is an important skill for higher programs in mathematics and science.
The word "Algebra" originates from the Persian mathematician Muhammad ibn Musa al-Khwarizmi, who wrote 'Kitab Al Muhtasar fi Hisab Al Jabr Wa’l Muqabala', later translated as 'The Compendious Book on Calculation by Completion and Balancing'. The term "Al-Jabr" in the title is the source of the word "Algebra." Al-Khwarizmi defined algebra as a method for reducing and balancing equations by rearranging terms and canceling alike parts to solve unknowns.
Long before Al-Khwarizmi, the ancient Mesopotamians and Egyptians developed early algebraic ideas using symbols for unknown quantities. Around 2000 BC, Mesopotamians recorded calculations on clay tablets, including tables of squares up to 59, which aided multiplication and other operations.
The ancient Egyptians tackled "aha" problems, or finding unknown values, as seen in the Moscow and Rhind papyri containing linear equations. Additionally, the Berlin papyrus shows their understanding of quadratic equations, illustrating the long and rich history of algebraic concepts.
Algebra is divided into various branches, which focus on different aspects. The different types of algebra are:
1. Pre-Algebra - It encompasses fundamental concepts that can help transform real-life situations into algebraic expressions.
Example: Vinny has 4 apples, and her friend has 12 more apples than her. Vinny and her friend have a total of 20 apples. How many apples does her friend have?
Solution: Given that Vinny has 4 apples, and the total number of apples is 20:
From the second condition, we have 4 + x = 20.
So, x = 20 – 4
x = 16
Thus, Vinny's friend has 16 apples.
2. Elementary Algebra - The branch of algebra that deals with basic operations, such as addition, subtraction, division, and multiplication.
Example: Solve the equation 2x + 6 = 18.
Subtract six from both sides of the equation.
2x + 6 – 6 = 18 – 6
2x = 12
x = 12 ÷ 2
x = 6
3. Abstract Algebra - The branch of algebra that deals with abstract concepts, such as fields, groups, and modules, is called abstract algebra.
Example: The 12-hour clock is an example of a cyclic group in abstract algebra. It tells us about how numbers return to the beginning after they reach their maximum value. This demonstrates the idea of the primary structure of modular arithmetic.
4. Universal Algebra - The branch of algebra that deals with common properties of all algebraic structures, like rings, fields, modules, lattices, etc.
Example: Boolean algebra is an example of Universal algebra. In Boolean algebra, there are logical operations like AND, OR, and NOT.
There are two binary operations in Boolean algebra. They are "∧" (AND) and "∨" (OR).
There is one unary operation:" ¬ " (complement or NOT).
There are two constants: 0 and 1.
Boolean algebra is an example of Universal algebra because it covers all structures and their common properties in algebra. Boolean algebra is also used to prove many theorems in mathematics.
Properties:
The following fundamental properties govern Boolean algebra, defining how logical variables interact under AND (∧), OR (∨), and NOT (¬) operations. These rules are essential for simplifying and analyzing logic expressions in mathematics, computer science, and digital circuit design.
Idempotence: Applying the same operation twice on a variable yields the same result as applying it once;
x ∧ x = x
x ∨ x = x
Commutativity: The order of variables does not affect the result of the operation.
x ∧ y = y ∧ x
x ∨ y = y ∨ x
Associativity: Grouping of variables does not change the outcome.
(x ∧ y) ∧ z = x ∧ (y ∧ z)
(x ∨ y) ∨ z = x ∨ (y ∨ z)
Absorption: Combining a variable with an operation involving itself and another variable simplifies to the original variable.
x ∧ (x ∨ y) = x
x ∨ (x ∧ y) = x
Distributivity: This property shows how one operation can be spread over another inside an expression. Specifically, the AND operation (∧) distributes over OR (∨), meaning:
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
Similarly, OR distributes over AND:
x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
This allows logical expressions to be rearranged and simplified.
Complement Laws: These laws describe how a variable interacts with its complement (negation). A variable ANDed with its complement is always false (0):
x ∧ ¬x = 0
While a variable ORed with its complement is always true (1):
x ∨ ¬x = 1
These express the fundamental idea that something cannot be both true and false simultaneously, and must be either true or false.
5. Linear Algebra - Linear algebra is the branch of algebra that deals with vectors, vector spaces (also known as linear spaces), and linear transformations between those spaces.
Example: Adding two vectors,
\(A = { \begin{bmatrix} 2 \\[0.3em] 3 \\[0.3em] \end{bmatrix}}\), \(B= { \begin{bmatrix} 5 \\[0.3em] 3 \\[0.3em] \end{bmatrix}}\)
\(A + B = { \begin{bmatrix} 2 + 5 \\[0.3em] 3 + 3 \\[0.3em] \end{bmatrix}} = { \begin{bmatrix} 7 \\[0.3em] 6 \\[0.3em] \end{bmatrix}} \)
6. Commutative Algebra - The branch of algebra that focuses on studying commutative rings, their ideals, and the structure built on them.
In simple terms, it examines systems in math where the order of addition or multiplication doesn't matter.
Example: Let us consider two integers a and b.
Commutativity of Addition: a + b = b + a
Commutativity of Multiplication: a × b = b × a
7. Advanced Algebra - Advanced Algebra is an extension of introductory algebra. It includes new topics that are essential for higher-level mathematical calculations. Advanced algebra is also referred to as Algebra 2.
Example: Polynomials, Rational Expressions, Quadratic Equations and Functions, Exponents and Logarithmic Functions, Conic Sections, etc.
Each branch of algebra has its own formulas and deals with solving distinct types of problems.
Algebraic expressions are the representation of variables and constants and operations like addition, subtraction, multiplication, and division.
Examples:
Why do we need equations in algebra? Variables, coefficients, constants, and mathematical operators (such as +, -, =, etc.) together form an equation. Through simple operations, we get the value of the variables. Equations in algebra come in various types, each with its own characteristics. These are the different types of equations in Algebra.
1. Linear Equations - An equation of degree one is known as a linear equation. The standard form of a linear equation is f(x) = ax + b, where a ≠ 0 and both a and b are constants while x is a variable of degree 1.
Example: \( 7x + 25 = 0, \quad 2x + 3y + 15 = 0 \)
2. Quadratic Equations - An equation of degree 2 is known as a quadratic equation. The standard form of a quadratic equation is f(x) = ax2 + bx + c (a ≠ 0) where a, b, and c are constants and x is a variable.
Example: \( 2x^2 + 3x + 14 = 0, \quad 8x^2 + 17x + 147 = 0 \)
3. Cubic Equations - An equation of degree 3 is known as a cubic equation. The standard form of the cubic equation is f(x) = ax3 + bx2 + cx + d, where a ≠ 0, and a, b, c, and d are constants and x is a variable.
Example: \( 6x^3 + 12x^2 + 35x + 40 = 0, \quad 32x^3 + 46x^2 + 55x + 127 = 0 \)
4. Polynomial Equations - Polynomial equations are equations that consist of variables, exponents, and coefficients. All polynomial equations are part of algebraic equations.
Example: \( 2x + 3y = 0, \quad 7x^3 + 12x^2 + 35x + 30 = 0 \)
5. Rational Equations - The equations that consist of at least one fraction, where both the numerator and denominator are polynomials, are called rational equations.
Example: \( {3 \over (x+1)} + {4 \over 3x} = 0 \)
6. Radical Equations - The equations that consist of radical forms such as square root, cube root, or any other type of root are called radical equations.
Example: \(\sqrt{ x +2} = 0, \sqrt[3]{4x} + \sqrt[4]{2y} = 0\)
7. Exponential Equations - The equations that have variables as exponents are called exponential equations.
Example: \( e^{2x} = 0, \quad 2^{3x} + 2 = 0 \)
8. Logarithmic Equations - The equation that consists of the logarithm of an expression containing a variable is called a logarithmic equation.
Example: \( \log(x) + \log(x-1) = 0 \)
9. Trigonometric Equations - Equations that consist of the expression containing the trigonometric functions of the variables are called trigonometric equations.
Example: \( \sin^{2}x + \cos^{3}x = 0 \)
Exponents: These are mathematical operations that are written in the form of an; a is the base and n is the power or the exponent. Exponents help us solve expressions.
It can be expressed in the form: an = ...n times.
Logarithms: The opposites of exponents are logarithms. Logarithms are used to solve complex algebraic formulas. The exponential form can be converted to logarithmic form by using the formula log an = x.
This includes the fundamental operations which are performed on variables and constants, similar to arithmetic operations. Given below are the operations in problem-solving in algebra.
Algebraic formulas are a combination of numbers and variables. Numbers are fixed when their value is known, and variables are used to represent unknown values.
Here are some examples of algebraic identities:
Algebraic operations are mathematical processes that involve the manipulation of numbers, symbols and variables to produce new expressions or results. Here are the basic properties of algebra.
1. Commutative Property - Commutative property states that changing the order of two numbers does not change the result. In other words, switching the operands still gives the same answer.
The arithmetic operators addition and multiplication satisfies the commutative property.
2. Associative Property - Associative property is a fundamental principle in mathematics that applies to operations like addition and multiplication. The result remains the same no matter how the numbers are grouped, as long as their order does not change.
Hence, it is proved.
3. Distributive Property - This property says that if you multiply a number by the sum or difference of two numbers, it’s the same as multiplying that number by each of the two numbers separately. Then, you add or subtract the results.
The form is a × (b + c) = a × b + a × c.
Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 14
4. Identity Property - Identity property is defined as the property where if any arithmetic operations are used to combine an identity with a number (n), the end result will be n.
5. Inverse Property - The Inverse Property in algebra refers to the concept that every number has an opposite or reciprocal that "undoes" the original operation, leading to an identity element.
6. Closure Property - The closure property states that if we perform operations like addition, multiplication, subtraction, and division on a set of numbers, then the result will also belong to the same set of numbers.
Let us consider integers 3 and 4. Performing addition on these integers will result in 3 + 4 = 7, which also belongs to the set of integers.
Here are some tips and tricks that can help children learn algebra in a simpler way. Children also get to know how variables, constants, and coefficients work in algebraic expressions.
Students can make mistakes in algebra as it involves many things like signs and variables. Let us see some common mistakes and learn how to avoid them.
Algebra may look like a collection of symbols and equations, but it teaches children how to think logically and solve problems step-by-step. It is a foundation for many practical tasks we come across in daily life. Here are some simple ways algebra is applied in everyday life:
Algebra helps in calculating saving goals. It can be used in budgeting and saving.
Measurement of ingredients in cooking and recipe adjustments.
Calculating how long the trip will take based on distance and speed. Example: Time= Distance/Speed.
Kinematic equations like v=u+at calculate an object’s velocity over time, which is important in learning about motion.
Used in economics for linear relationships in supply and demand.
Algebra is a vast field of advanced mathematics and has various career opportunities, so if you enjoy algebra, you know you’ve got another cooler option.
If (x + 5)² = 49, find the value of x.
Step 1: Identify values
Here, a = x and b = 5.
Step 2: Expand the square
(x + 5)2 = x2 + 52 + 2 × 5
Then, set equal to 49,
x2 + 10x + 25 = 49
Step 3: Move all terms to one side
x2 + 10x + 25 – 49 = 0
x2 + 10x - 24 = 0
Step 4: Split the middle term
x2 + 12x - 2x - 24 = 0
Step 5: Factor by grouping
(x + 12)(x - 2) = 0
Step 6: Solve for 𝑥
x = -12; x = 2.
By using identity (a + b)2 = a2 + b2 + 2ab
Verify if the identity 49 - 16 = (7 + 4)(7 - 4) holds true.
By using the identity,
a2 - b2 = (a + b)(a - b)
Here, a = 7 and b = 4
49 - 16 = 33 and (7 + 4)(7 - 4) = 11 × 3 = 33
Both sides are equal.
Find the value of (2 + 3)³ without directly calculating 5³.
We know that, a = 2 and b = 3
Now use the formula and calculate each term,
(2 + 3)3 = 23 + 3 × 22 × 3 + 3 × 2 × 32 + 33
Add all the terms
= 8 + 36 + 54 + 27 = 125
(2 + 3)3 = 125.
Using the identity,
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Solve x² - 5x + 6 = 0
x = 2 or x = 3.
We know that,
x² - 2x - 3x + 6 = 0
Factorize the equation
(x - 2)(x - 3) = 0
x - 2 = 0 or x - 3 = 0
Solve for x in the equation, 2x + 5 = 15
x = 5
We are solving the linear equation:
2x = 15 - 5
Simplify the right-hand side of the equation
2x = 10
x =
Now, solve for x and simplify the fraction
x = 5.
Algebra is more than just solving equations, students build a strong foundation for topics in science, engineering, and economics. Ultimately, algebra helps children develop a mindset of solving problems with ease and practice. Students will realize algebra is not just an academic subject, but a tool used in everyday life.
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Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.