Revisions to How to display very small numbers in Mathematica?

Tweak the MathJax. Reword.

edited Jan 21, 2019 at 0:19

5.3k
1
10
17

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: In this particular case, given that $x$ is a small number, $\,|x|<<1,\,$ then then we get $$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$$$\cos(x) \approx 1 - \frac{x^2}2, \quad e^{-c\,x} \approx 1 - cx,\quad \cos(x) - e^{-c\,x} \approx c\,x.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: In this particular case, given that $x$ is a small number, $\,|x|<<1,\,$ then we get $$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: In this particular case, given that $x$ is small, $\,|x|<<1,\,$ then we get $$\cos(x) \approx 1 - \frac{x^2}2, \quad e^{-c\,x} \approx 1 - cx,\quad \cos(x) - e^{-c\,x} \approx c\,x.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

Rewording.

Source Link

edited Jan 21, 2019 at 0:04

Somos

5.3k
1
10
17

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: You can easily find the answer in: In this particular case. Given, given that $x$ is a small number, $\,|x|<<1,\,$ then we get $$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: You can easily find the answer in this particular case. Given that $x$ is a small number, then $$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: In this particular case, given that $x$ is a small number, $\,|x|<<1,\,$ then we get $$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

Fixed a sign error.

Source Link

edited Jan 20, 2019 at 19:12

Somos

5.3k
1
10
17

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: You can easily find the answer in this particular case. Given that $x$ is a small number, then $$\cos(x) \approx 1 - x^2/2, \, \exp(cx) \approx 1 + cx,\,\cos(x) - \exp(cx) \approx cx.$$$$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: You can easily find the answer in this particular case. Given that $x$ is a small number, then $$\cos(x) \approx 1 - x^2/2, \, \exp(cx) \approx 1 + cx,\,\cos(x) - \exp(cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.

The simplest methods are usually the best. Try this code

N[Cos[x] - Exp[-x 27/10] /. x -> 17*^-26, 15] // InputForm

or a minor variation

With[{x = 17*^-26}, N[Cos[x] - Exp[-x 27/10], 15]] // InputForm

or define a function first

f = Cos[#] - Exp[-# 27/10] &; N[f[17*^-26], 15] // InputForm

All of these return the result

4.589999999999999999999998802094999`15.*^-25

You can get more digits by increasing the 15 digits precision.

For example, with 34 digits precision the result returned is

4.5899999999999999999999988020950000000000000000001613`34.*^-25

All those digits may be spurious since the given numbers $2.7$ and $1.7\times 10^{-25}$ seem to have only 2 digits of precision. In that case, the answer using 2 digits of precision is $4.6\times 10^{-25}$.

Note: You can easily find the answer in this particular case. Given that $x$ is a small number, then $$\cos(x) \approx 1 - x^2/2, \, \exp(-cx) \approx 1 - cx,\,\cos(x) - \exp(-cx) \approx cx.$$ The simple answer is thus $2.7 \cdot 1.7\times 10^{-25}$.