Finding the Real Solution for Fractional Part Equations with Given Values of k

  • Thread starter juantheron
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In summary, for the equation \{x\} = \{x^2\} = \{x^3\}, where \{x\} represents the fractional part of x, we can take \{x\} = \{x^2\} = \{x^3\} = k, where k is a value between 0 and 1. By setting x = \lfloor x \rfloor + k, where \lfloor x \rfloor is the floor function of x, we can solve for x. Further testing of values for k can help find the solutions to this equation.
  • #1
juantheron
247
1
Calculation of Real ##x## in ##\{x\} = \{x^2\} = \{x^3\}##, Where ##\{x\} = ## fractional part of ##x##

My Try:: We Know that ##\{x\} ## and ##\{x^2\}## and ##\{x^3\}## are are ##\in \left[0,1\right)##

So Let we take ##\{x\} = \{x^2\} = \{x^3\} = k##, where ##k\in \left[0,1\right)##

So If ##\{x\} = k## , Then ##x-\lfloor x \rfloor = k\Rightarrow x = \lfloor x \rfloor +k##

where ##\lfloor x \rfloor = ## floor function of ##x##

Similarly If ##\{x^2\} = k## , Then ##x^2-\lfloor x^2 \rfloor = k\Rightarrow x^2 = \lfloor x^2 \rfloor +k##

Sililarly If ##\{x^3\} = k## , Then ##x^3-\lfloor x \rfloor = k\Rightarrow x^3 = \lfloor x^3 \rfloor +k##

Now How Can I proceed after that,

please Help me

Thanks
 
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  • #2
Hint: test some values for k.
 

1. What is a fractional part equation?

A fractional part equation is an equation that involves a fraction, where the numerator and denominator are both algebraic expressions or numbers.

2. How do you solve a fractional part equation?

To solve a fractional part equation, you need to isolate the variable on one side of the equation and perform inverse operations to cancel out any terms on the other side. The goal is to get the variable alone on one side of the equation.

3. What are the common operations used in fractional part equations?

The common operations used in fractional part equations are addition, subtraction, multiplication, and division. These operations are used to manipulate the fractions and solve the equation.

4. Can a fractional part equation have more than one solution?

Yes, a fractional part equation can have multiple solutions. This is because a fraction can have different values depending on the values of the numerator and denominator, and there can be more than one combination of values that satisfy the equation.

5. How do fractional part equations relate to real-life situations?

Fractional part equations can be used to solve real-life problems involving fractions, such as calculating discounts, finding the final price after a discount, or determining the amount of ingredients needed for a recipe. They are also commonly used in financial and scientific calculations.

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