What is the value of arctanh(0.99) ?

The value of arctanh(0.99) is (ln(199))/2

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arctanh(0.99)

arctanh (0.99)

\frac{ln ( 199 )}{2}

Decimal Notation

2.64665 \dots

Solution steps

arctanh (0.99)

= arctanh (\frac{99}{100})

Rewrite using trig identities:

\frac{ln ( 199 )}{2}

arctanh (\frac{99}{100})

Use the Hyperbolic identity:

arctanh (x) = \frac{ln ( \frac{1 + x}{1 - x} )}{2}

= \frac{ln ( \frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} )}{2}

\frac{ln ( \frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} )}{2} = \frac{ln ( 199 )}{2}

\frac{ln ( \frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} )}{2}

\frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} = 199

\frac{1 + \frac{99}{100}}{1 - \frac{99}{100}}

Join

1 - \frac{99}{100} : \frac{1}{100}

1 - \frac{99}{100}

Convert element to fraction:

1 = \frac{1 \cdot 100}{100}

= \frac{1 \cdot 100}{100} - \frac{99}{100}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 100 - 99}{100}

1 \cdot 100 - 99 = 1

1 \cdot 100 - 99

Multiply the numbers:

1 \cdot 100 = 100

= 100 - 99

Subtract the numbers:

100 - 99 = 1

= 1

= \frac{1}{100}

= \frac{1 + \frac{99}{100}}{\frac{1}{100}}

Join

1 + \frac{99}{100} : \frac{199}{100}

1 + \frac{99}{100}

Convert element to fraction:

1 = \frac{1 \cdot 100}{100}

= \frac{1 \cdot 100}{100} + \frac{99}{100}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 100 + 99}{100}

1 \cdot 100 + 99 = 199

1 \cdot 100 + 99

Multiply the numbers:

1 \cdot 100 = 100

= 100 + 99

Add the numbers:

100 + 99 = 199

= 199

= \frac{199}{100}

= \frac{\frac{199}{100}}{\frac{1}{100}}

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{199 \cdot 100}{100 \cdot 1}

Refine

= \frac{199 \cdot 100}{100}

Cancel the common factor:

100

= 199

= \frac{ln ( 199 )}{2}

= \frac{ln ( 199 )}{2}

= \frac{ln ( 199 )}{2}

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