Understanding the Mathematical Equality: √3 / 3 1 / √3
Understanding the Mathematical Equality: √3 / 3 1 / √3
Often, students and math enthusiasts are puzzled by the seemingly contradictory mathematical statement that √3 / 3 equals 1 / √3. This article aims to clarify this equality by breaking it down into simpler and more understandable steps, using a combination of algebraic manipulation and mathematical reasoning.
Initial Misconception
The statement √3 ÷ 3 ≠ 1 ÷ √3 is incorrect. Let's explore why this is the case and how the equality can be proven.
Division Basics
When dividing two numbers, the result is both the quotient and the remainder. For instance, when 3 is divided by 3, the quotient is 1 and the remainder is 0. This means that the statement 1 ÷ √3 does not equate to √3 because √3 is an irrational number and cannot be simplified to a quotient of 1 with no remainder.
Step-by-Step Demonstration
The correct way to prove the equality √3 / 3 1 / √3 involves the concept of rationalizing the denominator. This technique helps in eliminating the square root in the denominator, making the expression more manageable and understandable.
Rationalizing the Denominator
Consider the fraction . To rationalize the denominator, multiply both the numerator and the denominator by , since .
[ frac{1}{sqrt{3}} times frac{sqrt{3}}{sqrt{3}} frac{sqrt{3}}{sqrt{3}^2} frac{sqrt{3}}{3} ]
Algebraic Manipulation
Let's take a step further with another algebraic approach:
Left-Hand Side (L.H.S.) Approach
Starting with the expression , we can rewrite it as:
[ 3^{1/2} times frac{1}{3} 3^{1/2} times frac{1}{3^{1/2} times 3^{1/2}} frac{1}{3^{1/2}} frac{1}{sqrt{3}} ]
This shows that
Right-Hand Side (R.H.S.) Approach
Similarly, let's consider the expression . Using the same principle of rationalizing the denominator, we multiply both the numerator and the denominator by :
[ frac{1}{sqrt{3}} times frac{sqrt{3}}{sqrt{3}} frac{sqrt{3}}{sqrt{3}^2} frac{sqrt{3}}{3} ]
This confirms the equality 1 ÷ √3 √3 / 3.
Conclusion
Therefore, it is clear that √3 / 3 is indeed equal to 1 / √3, as proven through algebraic manipulation and the process of rationalizing the denominator. This equality showcases the beauty and complexity of mathematics, confirming that seemingly contradictory expressions can be proven to be equal under the right mathematical operations.