Algebraic Expressions 1

Using the language of Algebra, we can represent certain real life situations in terms of what are called “algebraic expressions”. They may contain the following elements.

1. ‘x, y or z to represent variables

2. ‘a, b, c or numbers to express constants

3. Arithmetic operators to represent relations between them.

A few examples would clarify the idea.

1. What would be my age after 3 years?

   1. x + 3: My present age is not known and hence it is a variable. After 3 years, I would be 3 years older i.e present age + 3 years.

2. Because of Diwali, the prices of all refrigerators in Vivek & Co have been reduced by Rs 1000. What is the price of refrigerators now?

   1. y – 1000 where “y” is the present price of a particular refrigerator

3. What is area & perimeter of a rectangle?

   1. Area - xy: assuming x is the length and y is the breadth.

   2. Perimeter – 2x + 2y or 2(x + y)

4. If toffees are equally divided among 6 children, how many will each get, assuming that there are no remainders?

   1. x/6

5. My elder brother’s monthly pocket money was twice mine and in addition this month my uncle gave him Rs 10. How much money did he get this month?

   1. 2x + 10

6. Cost of 5 kgs of apples & 4 kgs of oranges

   1. 5x + 4y

7. Total weight of a box (which weighs 5 kgs) containing some clothes & books.

   1. x + y + 5

8. Area of a circle

   1. π where x is the radius

9. The temperature equivalent of centigrade expressed in Fahrenheit

   1. x + 32

10. The sum of the internal angles of a polygon (in degrees)

    1. 180*(N – 2) where N is the number of sides of the polygon.

We can see that many of the formulae we have learnt in the science & math classes are all examples of algebraic expressions.

Rhetoric Algebra to Symbolic Algebra

The above are also examples of Rhetoric Algebra converted to Symbolic Algebra. For many centuries algebra developed mainly as a discipline where the problem statement and the steps of the solution were expressed in terms of spoken sentences. Algebra was mainly seen as generalized arithmetic.

In many problems, the language itself becomes a barrier to understanding a problem. Converting these rhetoric statements into symbolic expressions, as has been done above, had many advantages.

The move towards symbolic algebra was started by Francois Viete in the 16thcentury. He helped replace the verbal method of doing algebra with the modern symbolism in which unknown variables are denoted by symbols like x, y, and z.

The advantages of symbolic algebra were many.

Firstly, all information which were not essential for solving the problem were left out. In example 5, the essential statement was that my brother got twice my allowance and Rs 10 more. This was represented as 2x + 10. This helps us to focus our attention on the core of the problem.

Secondly, the symbols can be manipulated as per rules of operation. The manipulation can be done without any reference to the original problem. So the steps of the solution can be written as an algorithm.