ALGEBRAIC LANGUAGE: CONCEPT, WHAT IT IS FOR, EXAMPLES, EXERCISES - MATHS - 2025

The algebraic language is the one that uses letters, symbols and numbers to express briefly and concisely sentences in which mathematical operations are required. For example 2x - x ² is algebraic language.

Using the appropriate algebraic language is very important to model many situations that occur in nature and in everyday life, some of which can be very complex depending on the number of variables that are handled.

Algebraic language consists of symbols, letters, and numbers that briefly express mathematical propositions. Source: Pixabay.

We are going to show some simple examples, for example the following: Express in algebraic language the phrase «Double a number».

The first thing to take into account is that we do not know how much that number is worth. Since there are many to choose from, then we are going to call it "x", which represents them all and then we multiply it by 2:

Double a number is equal to: 2x

Let's try this other proposition:

As we already know that we can call any unknown number "x", we multiply it by 3 and add the unit, which is nothing other than the number 1, like this:

The triple of a number plus unity equals: 3x + 1

Once we have the proposition translated into algebraic language, we can then give it the numerical value we want, to carry out operations such as addition, subtraction, multiplication, division and many more.

What is algebraic language for?

The immediate advantage of algebraic language is how short and concise it is. Once handled, the reader appreciates properties at a glance that would otherwise take many paragraphs to describe and some time to read.

In addition, being brief, it facilitates operations between expressions and propositions, especially when we use symbols such as =, x, +, -, to name a few of the many that mathematics has.

In short, an algebraic expression would be, for a proposition, the equivalent of looking at a photo of a landscape, instead of reading a long description in words. Therefore, the algebraic language facilitates analysis and operations and makes texts much shorter.

And that's not all, the algebraic language allows you to write general expressions, and then use them to find very specific things.

Suppose for example that we are asked to find the value of: "triple a number plus the unit when said number is worth 10".

Having the algebraic expression, it is easy to substitute "x" for 10 and carry out the operation described:

(3 × 10) + 1 = 31

If later we want to find the result with another value of "x", it can be done just as quickly.

A little history

Although we are familiar with mathematical letters and symbols such as the “=”, the letter “x” for the unknowns, the cross “x” for the product, and many others, these were not always used to write equations and sentences.

For example, ancient Arabic and Egyptian math texts contained hardly any symbols, and without them, we can already imagine how extensive they must have been.

However, it was the same Muslim mathematicians who began to develop the algebraic language from the Middle Ages. But it was the French mathematician and cryptographer François Viete (1540-1603) who was the first known to write an equation using letters and symbols.

Some time later, the English mathematician William Oughtred wrote a book that he published in 1631, where he made use of symbols such as the cross for the product and the proportional symbol ∝, which are still used today.

With the passing of time and the contribution of many scientists, all the symbols that are used today in schools, universities and different professional fields developed.

And it is that mathematics is present in the exact sciences, economics, administration, social sciences and many other areas.

Examples of algebraic language

Here are examples of using algebraic language, not just to express propositions in terms of symbols, letters, and numbers.

Figure 2.- Table with some commonly used propositions and their equivalent in algebraic language. Source: F. Zapata.

Sometimes we must go in the opposite direction, and having an algebraic expression, write it with words.

Note: although the use of the "x" as a symbol of the unknown is very widespread (the frequent "… find the value of x…" of the tests), the truth is that we can use any letter we want to express the value of some magnitude.

The important thing is to be consistent during the procedure.

- Example 1

Write the following sentences using algebraic language:

a) The quotient between the double of a number and the triple of the same plus the unit

Answer to

Let n be the unknown number. The expression searched is:

b) Five times a number plus 12 units:

Answer b

If m is the number, multiply by 5 and add 12:

c) The product of three consecutive natural numbers:

Answer c

Let x be one of the numbers, the natural number that follows is (x + 1) and the one that follows this is (x + 1 + 1) = x + 2. Therefore the product of the three is:

d) The sum of five consecutive natural numbers:

Answer d

Five consecutive natural numbers are:

Reply

Sometimes the phrase "… decreased by" is used to express a subtraction. In this way the previous expression would be:

Double a number diminished in its square.

Exercise resolved

The difference of two numbers is equal to 2. It is also known that 3 times the greater, added with twice the smaller, is equal to four times the aforementioned difference. How much is the sum of the numbers worth?

Solution

We will carefully analyze the situation presented. The first sentence tells us that there are two numbers, which we will call x and y.

One of them is larger, but it is not known which one, so we will assume it is x. And its difference is equal to 2, therefore we write:

x - y = 2

Then it is explained to us that "3 times the greatest…", this is equal to 3x. Then it goes: added with "twice the smallest…", which is equivalent to 2y… Let's pause and write here:

3x + 2y….

Now we continue: “… is equal to four times the aforementioned difference”. The aforementioned difference is 2 and we can now complete the proposition:

3x + 2y = 4.2 = 8

With these two propositions we have to find the sum of the numbers. But to add them we first have to know what they are.

We return to our two propositions:

x - y = 2

3x - 2y = 8

We can solve for x from the first equation: x = 2 + y. Then replace in the second:

3 (2 + y) - 2y = 8

y + 6 = 8

y = 2

With this result and substituting, x = 4 and what the problem asks for is the sum of both: 6.

References

1. Arellano, I. Brief history of mathematical symbols. Recovered from: cienciorama.unam.mx.
2. Baldor, A. 1974. Elementary Algebra. Cultural Venezolana SA
3. Jiménez, R. 2008. Algebra. Prentice Hall.
4. Méndez, A. 2009. Mathematics I. Editorial Santillana.
5. Zill, D. 1984. Algebra and Trigonometry. McGraw Hill.