Exploring Numbers That Add Up More Than They Multiply

The question of finding a number that, when added to itself, results in a larger value than when it is multiplied by itself, is a fascinating exercise in basic mathematics. This exploration can help us understand the properties of numbers and the subtleties that arise when comparing addition and multiplication. Let's delve into this interesting problem and find the solution.

Setting Up the Inequality

The inequality we need to solve is straightforward: find a number x such that 2x > x^2. This can be written mathematically as:

2x - x^2 > 0

By rearranging the terms, we get:

x^2 - 2x

This can be factored to:

x(x - 2)

Solving the Inequality

To solve this inequality, we need to determine the values of x for which the expression x(x - 2) . This expression is a quadratic and changes sign at the roots x 0 and x 2. We can analyze the intervals determined by these roots:

For x : Both factors are negative, so their product is positive ((-)(-) ). This does not satisfy the inequality. For 0 : One factor is positive and the other is negative, so their product is negative (( )(-) -). This satisfies the inequality. For x > 2: Both factors are positive, so their product is positive (( )( ) ). This does not satisfy the inequality.

Conclusion

The solution to the inequality 2x - x^2 > 0 is 0 . This means that any number within this range will satisfy the condition that adding it to itself results in a larger number than multiplying it by itself.

For example:

x 1 gives: Addition: 1 1 2 Multiplication: 1 * 1 1 2 > 1

Similarly, for any other number in the range 0 , the inequality will hold true.

Examples

Positive Fractions:

x 1/2 Addition: 1/2 1/2 1 Multiplication: 1/2 * 1/2 1/4 1 > 1/4

Negative Numbers:

Negative numbers will always produce a larger result when added to themselves than when multiplied by themselves, as shown by the example:

x -1 Addition: -1 -1 -2 Multiplication: -1 * -1 1 -2

Conclusion

In summary, the range of numbers that satisfy the condition of adding to themselves resulting in a larger value than multiplying by themselves is from 0 to 2, excluding the endpoints. This exploration not only helps us understand the behavior of numbers under addition and multiplication but also demonstrates the importance of considering the sign of the numbers involved.