Literal Calculations: First Expressions
Problem — How can we use letters to represent numbers and perform calculations in a general way?
- Understand what literal calculation is and why letters are used in mathematics.
- Be able to recognize and write simple literal expressions.
- Learn to perform operations on these expressions (addition, multiplication).
- Be capable of interpreting and manipulating these expressions in real-world problems.
Part 1: What is literal calculation?
Literal calculation is a branch of mathematics that uses letters to represent unknown or variable numbers in expressions and calculations.
In middle school, you've often worked with specific numbers like 5, 10, or 100. But sometimes, we need to reason with numbers that can change or that we don’t yet know. That’s where literal calculation comes in.
For example, in a recipe requiring "x" eggs, we write "2x" to mean "two times that number of eggs." Writing with letters allows us to generalize calculations and solve problems more easily.
Why use letters?
- To represent unknown numbers, called variables.
- To write formulas that adapt to different situations.
- To simplify calculations by grouping similar terms.
Literal calculation is used to manipulate expressions containing letters that represent numbers we do not know yet or that can vary. This allows us to reason more generally and solve more complex problems than those handled with numbers alone.
Part 2: The first literal expressions
A literal expression is a sequence of numbers, letters, and mathematical signs (+, -, ×, ÷) that represents a value or a calculation.
A literal expression can be as simple as "3x" or "x + 5." The letters, often called variables, can be replaced by a number to calculate the result.
Some examples of literal expressions:
- 3x: three times a number "x."
- x + 5: a number "x" increased by 5.
- 2a + 3b: the sum of twice a number "a" and three times a number "b."
To better understand, we can replace the letters by chosen numbers:
- If x = 4, then 3x = 3 × 4 = 12.
- If a = 2 and b = 3, then 2a + 3b = 2 × 2 + 3 × 3 = 4 + 9 = 13.
Literal expressions are combinations of numbers and letters that allow general calculations involving variables that can vary. Understanding how to read and interpret these expressions is an essential first step in literal calculation.
Part 3: Working with expressions: addition and multiplication
In literal calculation, letters follow the same rules as numbers for operations, but it is important to pay attention to units and terms that can be combined.
Here are the main operations on literal expressions covered here: addition and multiplication.
1) Addition of literal expressions
You can only directly add "like" terms—that is, terms with the same letter and the same power.
Example:
- 3x + 5x = (3 + 5)x = 8x
- 2a + 4b: cannot be added because the terms are different.
2) Multiplying by a number or a letter
Multiplying a literal expression by a number means multiplying each term by that number.
Example:
- 3 × (2x + 5) = 3 × 2x + 3 × 5 = 6x + 15
Multiplying letters by each other is written by placing the letters side by side:
x × a = xa. For example:
- 2x × 3a = 6xa
We learn to add only like terms in literal calculation and to multiply an expression by a number or a letter. These skills are the foundation for simplifying and transforming literal expressions, which is essential for solving problems later.
Part 4: Concrete examples of using literal calculation
Literal calculation is used to represent and solve various situations where quantities can change.
Example 1: The perimeter of a rectangle
Consider a rectangle with length l and width L. Its perimeter P is the sum of all its sides:
P = l + L + l + L = 2l + 2L
This gives a literal expression of the perimeter, which can be used to calculate P for different dimensions.
Example 2: Twice the sum
Calculating twice the sum of two numbers a and b is written as:
2 × (a + b) = 2a + 2b
Example 3: A numeric problem
A gardener plants x rows of flowers, with 5 flowers per row. The total number of flowers is then:
5 × x = 5x
If the gardener plants 8 rows, then the number of flowers is:
5 × 8 = 40
Literal expressions allow modeling concrete situations using letters that represent variable quantities. These models help generalize calculations and solve problems encountered in everyday life or in science.
Literal calculation introduces a powerful mathematical language: expressions with letters. We have seen how to recognize these expressions, write them, add or multiply them, and use them to model concrete situations. Mastering these basics is essential for progressing in mathematics, as they allow reasoning in a general way, anticipating, and solving various problems. This course thus lays the necessary foundations to approach more advanced concepts of literal calculation in the following years.