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Algebra is a branch of Mathematics that involves using letters and a set of rules or formulas to determine unknown values. Since letters are used to represent numbers in Algebra, all the mathematical operations that are generally performed on numbers can be performed on letters as well. Symbols in Algebra are used to show the relationship between the different quantities in an expression. Variables, expressions, and equations are the basic building blocks of Algebra. The value of the variable is not fixed and it can take different values.
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Key Terms: Algebra, Algebra Symbols, Symbols, Variables, Expressions, Equation, Commutativity, Order of operations, Basic Algebra, Mathematical Operations
Variables
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Variables are letters that are used to represent numbers and whose values may change depending on the situation.
For example, to form a ‘T’ using matchsticks, two matchsticks are needed. To form 2 ‘T’s, 4 matchsticks are needed.
Similarly, the number of matchsticks needed will vary depending on the number of ‘T’s being formed. So, we can infer the rule that
The number of matchsticks required = 2n where n is the number of times the process is to be repeated.
Using this rule we can calculate the number of matchsticks required to form 5 ‘T’s as
2 × n = 2 × 5 = 10
- Variables are used in many geometric formulas to calculate areas and perimeters of different shapes. For example, consider a square with the length of the side ‘s’. To calculate the perimeter of the square we need to add the length of the four sides.
Perimeter of the square = s + s + s + s = 4 × s
Here, the perimeter of the square depends on the value of the variable ‘s’.
- Variables are used in different arithmetic rules like commutativity and distributivity which are very important for certain calculations. For example, the commutative property of the addition of two numbers refers to the addition of the two whole numbers by interchanging them. The sum of the two numbers is the same in either case.
For example, 5 + 3 = 3 + 5 = 8. This property can be represented using variables as
x + y = y + x, where x and y are variables.
This property holds true for the multiplication of whole numbers as well.
Expressions
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Expressions can be defined as a combination of variables and arithmetic operators or symbols. Even though variables are represented as letters, it is important to remember that they are actually numbers. Hence, all the operations like addition, subtraction, multiplication, and division can be performed on the variables. Expressions cannot be evaluated unless a value is assigned to the variable in the expression.
For example, consider an expression 2x + 3 where x is a variable. Unless some value is assigned to x, the value of the expression 2x + 3 cannot be determined.
If x = 4, then the value of the expression will be
2x + 3
= 2(4) + 3
= 8 + 3
= 11
Equations
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An equation is a statement of equality that serves as a condition on a variable that holds true for a particular value of the variable. An equation has an equal to ( = ) sign that acts as a bridge between the Left Hand Side (LHS) expression and the Right Hand Side (RHS) expression.
- For example, consider the equation 4x + 5 = 25.
To find the value of the variable x,
4x + 5 = 25
4x = 25 - 5
4x = 20
x = 5
Here the value of variable x that satisfies the equation is 5 which is called the solution of the equation.
Symbols
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There are several symbols in Algebra that are used in equations and expressions to depict the relationship between the different variables. Symbols in Algebra make understanding complex equations and expressions easier. There are more than 10,000 symbols in Algebra. Some of the common algebra symbols are tabulated below:
| Symbol | Symbol Name | Example |
|---|---|---|
| = | Equal to | x=6 |
| + | Add | 7 + 4 = 11 |
| - | Subtract | 7 - 4 = 3 |
| ≡ | Identically Equal to | 7 ≡ 7 |
| ≠ | Not Equal to | 7 + 4 ≠ 4 |
| × | Multiply | 5 × 2 = 10 |
| ÷ | Divide | 30 ÷ 5 = 6 |
| > | Greater than | 24 > 20 |
| < | Less than | 18 < 20 |
| ≥ | Greater than or Equal to | a + b ≥ c |
| ≤ | Less than or Equal to | a + b ≤ c |
| % | Percentage | 30% |
| √ | Square Root | √100 = 10 |
| â | Cube Root | â1331 = 11 |
| . | Decimal Point | 13.45 |
| ≈ | Approximately equal to | x ≈ 0.5 |
| ∈ | Belongs to | 5 ∈ Natural Numbers |
| ∉ | Does not belong to | 0 ∉ Natural Numbers |
| ∞ | Infinity | Natural numbers are 1, 2, 3, ……..,∞ |
| ∝ | Proportional | x∝y x= ky |
| () | Parentheses | 5 - (2 + 1) = 5 - 3 = 2 |
| [] | Square brackets | 10 × [5 + (4 - 2)] = 10 × [5 + 2] = 10 × 7 = 70 |
| {} | Curly braces | 20 ÷ { 2 × [3 + (4 - 2)]} = 20 ÷ {2 × [3 + 2]} = 20 ÷ {2 × 5} = 20 ÷ 10 = 2 |
| ! | Factorial | 3! = 3 × 2 × 1 = 6 |
| R | Real Numbers | 4 ∈ R |
| Z | Integers | 9 ∈ Z |
| N | Natural Numbers | 5 ∈ N |
| Q | Rational Numbers | â, â are rational numbers as denominator not equal to 0 |
| P | Irrational Numbers | √2, √3 are irrational numbers |
| ⇒ | Implies | x- 2 = 5 â¹ x = 7 |
Things to Remember
- Algebra symbols are used to show the relationship between the variables in an expression or an equation.
- Variables are used to represent numbers and can take different values from a relevant collection of values.
- Expressions are a combination of variables and symbols that cannot be evaluated unless a value is assigned to the variable
- An Equation is an equality statement consisting of variables, symbols, and expressions.
- Some of the basic algebraic symbols are +, -, ×, ÷, =, ≠, ≤, ≥, <, >.
Sample Questions
Ques. Mention two geometric formulas where variables are used. ( 1 Marks)
Ans. The geometric formulas where variables are used are:
Perimeter of a square = 4 × s, where s is a variable
Perimeter of a rectangle = 2 × l + 2 × b, where l and b are variables.
Ques. Write the rule to find the number of matchsticks required to form a pattern of the letter ‘E’ using variables. ( 1 Marks)
Ans. The number of matchsticks required to form a pattern of the letter ‘E’ = 4n where n is the variable representing the number of ‘E’s to be formed.
Ques. The teacher distributes 5 pencils to each student. Find the number of pencils needed if the number of students is given as s. ( 1 Marks)
Ans.
Number of pencils given to each student = 5
Number of students = s
Total number of pencils required = 5 × s
Ques. The length of a rectangular box is 4cm less than 5 times the breadth of the box. If the breadth of the box is b cm, what is the length of the box? ( 1 Marks)
Ans.
Breadth of the box = b cm
Length of the box = 5(breadth of box) - 4
` = 5b - 4
Ques. What are the algebraic symbols used for comparing two variables or numbers?( 1 Marks)
Ans.
The algebraic symbols used for comparing two variables or numbers are <, >, =, ≠, ≤, ≥, ≡, â£.
Ques. How many symbols are used in Algebra? ( 1 Marks)
Ans.
There are more than 10,000 symbols in Algebra. Some of the symbols are rarely used.
Ques. What are the symbols used to represent different sets of numbers? ( 1 Marks)
Ans. The symbols used to represent different sets of numbers are
- N - Natural Numbers
- R - Real Numbers
- Z - Integers
- Q - Rational Numbers
- P - Irrational Numbers
Ques. What are the basic arithmetic symbols? ( 1 Marks)
Ans. The basic arithmetic symbols are:
- Plus sign (+) used for addition
- Minus sign (-) used for subtraction
- Multiply sign (×) used for multiplication
- Divide sign (÷) used for division.
Ques. Mention the arithmetic rules using variables.( 1 Marks)
Ans. The arithmetic rules expressed using variables:
- Commutativity of addition of two numbers can be expressed as x + y = y + x where x and y are variables.
- Distributivity of multiplication over addition can be expressed as
x × (y + z) = x × y + x × z.
Ques. List the different brackets used in algebra.
Ans. The different brackets used in Algebra are as follows:
- () - Parentheses
- [] - Square Brackets
- {} - Curly Braces
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