Constant and Variable

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Mathematics is a subject that deals with numbers, shapes, logic, quantity, and arrangements. There are two main branches of mathematics – pure mathematics and applied mathematics. Pure mathematics is an abstract science of numbers, quantity, and space, either as abstract concepts (pure mathematics) or as applied to other disciplines such as physics and engineering(applied mathematics).

Algebra, a part of pure mathematics is the concept based on unknown values called variables. The word “algebra” is derived from the Arabic word al-jabr, and this comes from the treatise written in the year $830$ by the medieval Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī.

The important concept of algebra is equations and it follows various rules to perform arithmetic operations. Let’s understand what is algebra and its two important concepts – constant and variable.

What is Algebra in Maths?

Algebra is a branch of mathematics that deals with symbols and the arithmetic operations operated on these symbols. These symbols do not have any fixed values and are called variables.

In our real-life problems, we often come across certain values that keep on changing and certain other values that remain constant(or fixed).

In algebra, the values that keep changing or can change are often represented with symbols such as $x$, $y$, $z$, $a$, $b$, or $c$. These symbols are called variables.

Further, these symbols are manipulated through various arithmetic operations of addition, subtraction, multiplication, and division, with the objective to find their values under certain given situations.

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Constant and variable are the two types of symbols used in algebra. They are a part of algebra and are considered the basic building blocks of algebraic expressions. A symbol that has a fixed numerical value is called a constant, whereas a quantity that has no fixed value but takes no various numerical values is called a variable.

Let’s understand these two terms in detail.

What are Constants?

Constants in algebra are that part of the algebraic expression that involves only numbers such as $5$, $-2$, $4.87$, $\frac{2}{3}$, $\sqrt{7}$, etc. We call them constants because their value always remains the same. It is fixed and does not change from one problem to another.

The number $6$ is a constant because it has a particular value, and it is known to everyone and it refers to the same value under different scenarios or problems. It can’t be changed.

Examples

Ex 1: Identify the constants in the algebraic expression $9x – 3$.

The constants in the algebraic expression $9x – 3$ are $9$ and $-3$.

Ex 2: Identify the constants in the algebraic expression $\frac {5x + 7}{2}$.

$\frac {5x + 7}{2} = \frac{5}{2} x + \frac{7}{2}$.

Therefore, the constants in the algebraic expression $\frac {5x + 7}{2}$ are $\frac{5}{2}$ and $\frac{7}{2}$.

Ex 3: Identify the constants in the algebraic expression $x – 5$.

$x – 5$ can be written as $1 \times x – 5$.

Therefore, the constants in the algebraic expression $x – 5$ are $1$ and $-5$.

What are Variables?

In Algebra, the letters represent variables. They are not constant, i.e., their value can be changed from time to time. A variable is represented by English letter, such as $x$, $y$, $z$, $p$, $q$, $r$, etc. Each of these symbols can take any numeric value depending on a problem or a situation where they are used.

Examples

Ex 1: Identify the variables in the algebraic expression $2x + 8$.

The variable in the algebraic expression $2x + 8$ is $x$.

Ex 2: Identify the variables in the algebraic expression $3a – 2b$.

The variables in the algebraic expression $3a – 2b$ are $a$ and $b$.

Ex 3: Identify the variables in the algebraic expression $x^{2} – 2xy + y^{2}$.

The variables in the algebraic expression $x^{2} – 2xy + y^{2}$ are $x$ and $y$.

Difference Between Variable and Constant

Following are the difference between variable and constant in algebra.

Signs Used in Algebra

One can perform mathematical operations like addition $\left(+ \right)$, subtraction $\left(− \right)$, multiplication $\left(\times \right)$ and division $\left( \div \right)$ in algebra to form meaningful algebraic expressions.

Apart from these four signs, there are some other signs and symbols that are also frequently used in algebra. These are equal to $\left(= \right)$, not equal to $\left(\ne \right)$, less than $\left(\lt \right)$, greater than $\left(\gt \right)$, less than equal to $\left(\le \right)$ and greater than equal to $\left( \ge \right)$.

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Algebra as Patterns

To understand the relationship between patterns and algebra, we need to try making some patterns. We can use matchsticks or crayons to construct a simple pattern and understand how to create a general expression to describe the entire pattern.

There are different types of algebraic patterns such as repeating patterns, growth patterns, number patterns, etc. All these patterns can be defined using different techniques.

Examples

Let’s go through some algebra patterns using matchsticks.

Ex 1: Number pattern $4$, $7$, $10$, $13$, … using matchsticks.

Number of matchsticks in $1^{st}$ shape = $4$.

Number of matchsticks in $2^{nd}$ shape = $7 = 4 + 3$.

Number of matchsticks in $3^{rd}$ shape = $10 = 7 + 3$.

Number of matchsticks in $4^{th}$ shape = $13 = 10 + 3$.

The pattern formed is an arithmetic pattern, where the first term $a = 4$ and the common difference $d = 3$.

Ex 2: Number pattern $3$, $5$, $7$, $9$, … using matchsticks.

Number of matchsticks in $1^{st}$ shape = $3$.

Number of matchsticks in $2^{nd}$ shape = $5 = 3 + 2$.

Number of matchsticks in $3^{rd}$ shape = $7 = 5 + 2$.

Number of matchsticks in $4^{th}$ shape = $9 = 7 + 2$.

Forming a Rule of Pattern in Algebraic Form

As seen in the above example, for creating a pattern, a certain set of rules needs to be considered. For applying these rules, we should first understand the nature of the sequence and the difference between the two consecutive numbers given in the pattern.

This rule can be expressed in algebraic form using variables, constants, and mathematical operators. A variable is used to denote a change in the pattern, whereas a constant is used to denote a constant factor of change and operator that is operated on constants and variables.

Let’s consider some examples to understand it.

Examples

Ex 1: Number pattern $4$, $7$, $10$, $13$, … using algebraic expression.

$1^{st}$ number = $4$

$2^{nd}$ number = $4 + 3 = 7$

$3^{rd}$ number = $7 + 3 = 10$

$4^{th}$ number = $10 + 3 = 13$

Here, the changing numbers are $4$, $7$, $10$, $13$, … and the fixed number is $3$, therefore, the numbers $4$, $7$, $10$, $13$, … can be represented by a variable such as $x$ and $3$ a constant.

Thus, the algebraic expression for $4$, $7$, $10$, $13$, … is $x + 3$.

Ex 2: Number pattern $3$, $5$, $7$, $9$, … using algebraic expression.

$1^{st}$ number = $3$

$2^{nd}$ number = $3 + 2 = 5$

$3^{rd}$ number = $5 + 2 = 7$

$4^{th}$ number = $7 + 2 = 9$

Here, the changing numbers are $3$, $5$, $7$, $9$, … and the fixed number is $2$, therefore, the numbers $3$, $5$, $7$, $9$, … can be represented by a variable such as $x$ and $2$ a constant.

Thus, the algebraic expression for $3$, $5$, $7$, $9$, … is $x + 2$.

Practice Problems

Form an algebraic expression for the following number patterns

$6$, $15$, $24$, $33$, $42$, …
$33$, $38$, $43$, $48$, $53$, …
$76$, $72$, $68$, $64$, $60$, …
$3$, $12$, $48$, $192$, …
$5$, $15$, $45$, $135$, …

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