How to Prove Mathematical Statements Without a Calculator

Proving mathematical statements without a calculator is a fundamental skill that enhances your understanding of numbers and their properties. In this article, we will explore a simple yet elegant proof involving negative and fractional powers, and demonstrate how to approach similar problems systematically.

The Proof: A Step-by-Step Breakdown

Consider the following expression:

(-0.125) 7 (1/2 * 1/3 * 1/4)

We want to prove that this expression is less than 10.

Negative and Fractional Powers

Let's break down the two parts of the expression separately:

Part 1: Negative Number to an Odd Power

A negative number raised to an odd power remains negative. Therefore, (-0.125) 7 is a small negative number. For context, 0.125 is equal to 1/8, and any power of 1/8 will result in a smaller fraction. Raising a fractional positive number to an odd power makes it even smaller.

Part 2: Fraction Times Fraction Times Fraction

The product (1/2 * 1/3 * 1/4) represents the multiplication of three smaller fractions, which equals 1/24. This fraction is very small compared to 1.

Combining the Parts

Now, let's combine the two parts:

(-0.125) 7 is a small negative fraction. (1/2 * 1/3 * 1/4) 1/24, which is a small positive fraction.

Add these two fractions:

A small negative fraction plus a small positive fraction results in a very small fraction. This small fraction can be positive or negative, but its magnitude is so small that it can be ignored for the purpose of this proof.

The Parentheses: Simplifying to a Positive Number Less Than 10

Now, let's look at the expression inside the parentheses:

9 1/3

This simplifies to:

9 0.3333... ≈ 9.3333... (or 9 and a third)

This positive number is less than 10.

Multiplying by a Fraction

Multiply this number by a fraction to get a smaller number. Here, we multiply 9.3333... by (1/2 * 1/3 * 1/4) 1/24:

9.3333... * (1/24) 9.3333... / 24 ≈ 0.3888...

This result is much smaller than 10.

The Conclusion: Right Side of the Equation

On the right side of the equation, we have:

1 * 10 10

This is exactly 10.

Final Comparison

The left side of the equation is some number less than 10, while the right side is exactly 10.

Oh gosh! That's what we wanted to prove! How wild is that!

Conclusion

As a proud witness to this elegant proof, let's sign off with:

Q.E.D.

In conclusion, by breaking down the components of the expression and understanding the properties of negative and fractional powers, we were able to prove that the given expression is less than 10 without a calculator. This approach can be applied to similar problems involving negative and fractional powers, and it enhances your ability to think critically about mathematical statements.