68/69 Additive Inverse :

The additive inverse of 68/69 is -68/69.

This means that when we add 68/69 and -68/69, the result is zero:

68/69 + (-68/69) = 0

Additive Inverse of a Fraction

For fractions, the additive inverse is found by negating the numerator or denominator, but not both. In this case:

  • Original fraction: 68/69
  • Additive inverse: -68/69

To verify: 68/69 + (-68/69) = 0

Extended Mathematical Exploration of 68/69

Let's explore various mathematical operations and concepts related to 68/69 and its additive inverse -68/69.

Basic Operations and Properties

  • Square of 68/69: 0.97122453266121
  • Cube of 68/69: 0.95714881479655
  • Square root of |68/69|: 0.99272717620543
  • Reciprocal of 68/69: 1.0147058823529
  • Double of 68/69: 1.9710144927536
  • Half of 68/69: 0.49275362318841
  • Absolute value of 68/69: 0.98550724637681

Trigonometric Functions

  • Sine of 68/69: 0.83355242102534
  • Cosine of 68/69: 0.55244036909226
  • Tangent of 68/69: 1.5088550143339

Exponential and Logarithmic Functions

  • e^68/69: 2.6791705393545
  • Natural log of 68/69: -0.014598799421153

Floor and Ceiling Functions

  • Floor of 68/69: 0
  • Ceiling of 68/69: 1

Interesting Properties and Relationships

  • The sum of 68/69 and its additive inverse (-68/69) is always 0.
  • The product of 68/69 and its additive inverse is: -4624
  • The average of 68/69 and its additive inverse is always 0.
  • The distance between 68/69 and its additive inverse on a number line is: 136

Applications in Algebra

Consider the equation: x + 68/69 = 0

The solution to this equation is x = -68/69, which is the additive inverse of 68/69.

Graphical Representation

On a coordinate plane:

  • The point (68/69, 0) is reflected across the y-axis to (-68/69, 0).
  • The midpoint between these two points is always (0, 0).

Series Involving 68/69 and Its Additive Inverse

Consider the alternating series: 68/69 + (-68/69) + 68/69 + (-68/69) + ...

The sum of this series oscillates between 0 and 68/69, never converging unless 68/69 is 0.

In Number Theory

For integer values:

  • If 68/69 is even, its additive inverse is also even.
  • If 68/69 is odd, its additive inverse is also odd.
  • The sum of the digits of 68/69 and its additive inverse may or may not be the same.

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