What is the golden rule of algebra?

The golden rule of algebra, often referred to as the "Golden Rule of Equations," is: "Do unto one side of the equation what you do to the other side." This means that when you perform an operation (such as addition, subtraction, multiplication, or division) on one side of an algebraic equation, you must do the same operation to the other side to maintain the equality of the equation. This rule ensures that the equation remains balanced and true.

Who is the father of algebra?

The title "Father of Algebra" is often attributed to the Persian mathematician Muhammad ibn Musa al-Khwarizmi. Al-Khwarizmi's work, particularly his book "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing), laid the foundations for algebra as a branch of mathematics.

what is boolean algebra?

Boolean algebra is a mathematical system used for working with binary variables, where variables can only have two possible values, typically represented as 0 and 1. In Boolean algebra, there are several rules and operations that help manipulate and simplify logical expressions. Some of the fundamental rules of Boolean algebra include: Identity Law: A + 0 = A A * 1 = A Null Law: A + A' = 1 A * A' = 0 Domination Law: A + 1 = 1 A * 0 = 0 Idempotent Law: A + A = A A * A = A Complement Law: A + A' = 1 A * A' = 0 Double Negation Law: (A')' = A Commutative Law: A + B = B + A A * B = B * A Associative Law: (A + B) + C = A + (B + C) (A * B) * C = A * (B * C) Distributive Law: A * (B + C) = (A * B) + (A * C) A + (B * C) = (A + B) * (A + C) De Morgan's Law: (A + B)' = A' * B' (A * B)' = A' + B' Absorption Law: A + (A * B) = A A * (A + B) = A Consensus Theorem: A * B + A' * C + B * C = A * B + A' * C These rules are essential for simplifying and manipulating logical expressions in Boolean algebra. They are widely used in digital logic design, computer science, and electronic circuit design to analyse and design logic circuits and systems.

Rules Of Algebra: Basics Of Algebra

Algebra is the branch of Maths that uses alphabetical letters to find unknown numbers. These letters are also called variables in algebra. The values which are known in the given expression such as numbers are called constants. When we speak about algebra basics, we refer to the general algebraic expressions, formulas, and identities, which are used to solve many mathematical problems. Let us learn the underlying concept of algebra with the help of some terminology, formulas, rules, examples, and solved problems.

In algebra, we begin to study math equations that have missing pieces, and use various skills to fill in those blanks.

When we start to learn math, we begin by learning how to count. Once we have mastered that, we learn the basic operations of arithmetic: addition, subtraction, multiplication, and division. Most students of a certain age can answer very quickly that $5 + 3 = 8$ or $2 \times 9 = 18$ .

However, what about $◻ \times 3 = 15$ ? What goes in the box? It has to be 5, right? Because 5 times 3 is 15. This is the beginning of the study of algebra.

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Know the terms of your algebraic equation. | Image source: Math Score

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Basics Of Algebra: Rules & Examples

A variable is a letter that is used in algebra to take the place of a number. Often we will use either n or x, but other letters can be used as well. Whenever we see a letter in algebra, it just means that a number of goes here, and we don't know what it is yet.

Here's an example.

$x - 4 = 2$

So, there is some number x (we don't know what it is yet), and when we subtract 4 from it, we get 2. What number is x? It's fun to think of algebra problems like a puzzle. With a little thinking, we can work out that x must be 6 because 6 - 4 = 2.

The basic parts of an algebra problem are its terms. Terms can be constants, or they can contain variables and coefficients. A constant term is just a number. A variable term contains a variable or a letter that takes the place of something we don't know yet and may have a coefficient or a number multiplying a variable. Terms are always separated by addition and subtraction.

In algebra, we can encounter expressions and equations. An expression is a statement in math that contains terms (constants, variables, and/or coefficients) separated by addition or subtraction and has no equals sign (or inequality like less than, greater than, etc.)

An equation is two or more expressions that are equal to each other, on either side of an equals sign. So, the key difference between an expression and an equation is that an equation has an equal sign, and an expression does not. Equations can be solved - that is, we can do algebra and figure out what the unknown variable has to be. Expressions can only be evaluated - if they contain a variable, we have to plug a given value into that variable and do the arithmetic.

Example

Algebra is important because we can use it to model real-world situations. For example, Priya goes to the game store and buys a bundle of 5 used video games on sale. When she gets home, she counts that she has 16 video games in her collection. How many video games did she have before she went to the game store?

To model this as an algebra equation, let's state the problem mathematically. Priya had some unknown number of games. We'll call this number x. Then, she added 5 to her collection, and afterward, she had 16 games. In mathematical notation, we write:

$x + 5 = 16$

Can we figure out what x must be? With a little bit of thought, we can see that x must be 11, because 11 + 5 = 16.

However, algebra isn't just about guessing and checking. There are some rules and skills we can use to solve algebra equations.

For Instance:

Algebraic equation.

x³ - 4x² + 3x + 5x² - 8x + 3x³ - 5 = 0

Solution:

To simplify the given algebraic equation, you can combine like terms by adding or subtracting them.

The equation is:

x^3 - 4x^2 + 3x + 5x^2 - 8x + 3x^3 - 5 = 0

First, let's combine the terms with the same exponent:

(x³ + 3x³) - (4x² - 5x²) + (3x - 8x) - 5 = 0

Now, simplify each group of like terms:

4x³ - x² - 5x - 5 = 0

So, the simplified form of the given equation is:

4x³ - x² - 5x - 5 = 0

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Learn and memorize all the basic algebraic formulas. | Image source: Toppr

Basic Formulas of Algebra

Algebra includes both numbers and letters. Numbers are fixed, i.e. their value is known. Letters or alphabets are used to represent unknown quantities in the algebra formula. Now, a combination of numbers, letters, factorials, matrices, etc. is used to form an equation or formula. This is essentially the methodology for algebra.

Here is a list of the basic algebraic formulas:

a² – b² = (a-b) (a+b)

(a+b)² = a² + 2ab + b²

(a-b)² = a² – 2ab + b²

a² + b² = (a-b)² +2ab

(a+b+c)² = a²+b²+c²+2ab+2ac+2bc

(a-b-c)² = a²+b²+c²-2ab-2ac+2bc

a³-b³ = (a-b) (a² + ab + b²)

a³+b³ = (a+b) (a² – ab + b²)

(a+b)³ = a³+ 3a²b + 3ab² + b³

(a-b)³ = a³- 3a²b + 3ab² – b³

“n” is a natural number, an – bn = (a-b) (an-1 + an-2b +….bn-2a + bn-1)

“n” is a even number, an + bn = (a+b) (an-1 – an-2b +….+ bn-2a – bn-1)

“n” is an odd number an + bn = (a-b) (an-1 – an-2b +…. – bn-2a + bn-1)

(am)(an) = am+n (ab)m = amn

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Basic Rules of Algebra

There are five basic rules in algebra.

Commutative Rule of Addition
Commutative Rule of Multiplication
Associative Rule of Addition
Associative Rule of Multiplication
Distributive Rule of Multiplication

Commutative Rule of Addition

In algebra, the commutative rule of addition states that when two terms are added, the order of addition does not matter. The equation for the same is written as, (a + b) = (b + a). For example, (x3 + 2x) = (2x + x3).

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Commutative Rule of Multiplication

The commutative rule of multiplication states that when two terms are multiplied, the order of multiplication does not matter. The equation for the same is written as, (a × b) = (b × a). For example,

(x4 - 2x) × 3x = 3x × (x4 - 2x).
LHS = (x4 - 2x) × 3x = (3x5 - 6x2)
RHS = 3x × (x4 - 2x) = (3x5 - 6x2)
Here, LHS = RHS, which proves that their values are equal.

Associative Rule of Addition

In algebra, the associative rule of addition states that when three or more terms are added, the order of addition does not matter. The equation for the same is written as, a + (b + c) = (a + b) + c. For example, x5 + (3x2 + 2) = (x5 + 3x2) + 2.

abide by laws — Rules are the guiding principles of algebraic operations. | Source: Gerd Altmann from Pixabay

Associative Rule of Multiplication

Similarly, the associative rule of multiplication states that when three or more terms are multiplied, the order of multiplication does not matter. The equation for the same is written as, a × (b × c) = (a × b) × c. For example, x3 × (2x4 × x) = (x3 × 2x4) × x.

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Distributive Rule of Multiplication

The distributive rule of multiplication states that when we multiply a number by the sum of two numbers, it results in the output which is the same as the sum of their products with the number individually. This is the distribution of multiplication over addition. The equation for the same is written as, a × (b + c) = (a × b) + (a × c). For example, x2 × (2x + 1) = (x2 × 2x) + (x2× 1).

The cardinal rule of algebra itself is balance. An equation has an equals sign, and whatever is on one side of the equals sign must equal what is on the other side of the equals sign. With that in mind, we can do anything we want to an equation - as long as we preserve the balance on both sides of the equals sign.

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Shreyanjana

Shreyanjana is an archaeologist who ironically finds the written word to be the most powerful means of storytelling. A travel buff and a photography enthusiast, she has been writing and sharing stories of all sorts ever since she can remember.

Rules Of Algebra: Formulas, Rules & Expressions