Real numbers that cause trouble with computers or are significant to computers.

close to 0.0

-0.0

close to 0.3

0.30000000000000004

related: 0.30000001192092896

close to 0.5

\[\frac{467807924713440738696537864469}{935615849440640907310521750000} \approx 0.499999999992646\]

0.666910

\[\frac{8391667}{12582905} \approx 0.666910\]

Handbook of Floating-Point Arithmetic

The divider of the first version of the Intel Pentium processor, released in 1994, was flawed [290, 122]. In extremely rare cases, one would get three correct decimal digits only. For instance, the computation of 8391667/12582905 would give 0.666869… instead of 0.666910… .

close to 1.0

\[\frac{\pi^4 + \pi^5}{e^6} \approx 0.9999999561918942\]

14

He tells us there are fourteen doors to his ‘house’, but a footnote informs us that, when Asterion uses the word ‘fourteen’, he means ‘infinite’

https://interestingliterature.com/2022/06/borges-house-of-asterion-summary-analysis/

16

INFINITY = 16

https://twitter.com/calebstanford4/status/1516537768175882246

close to 20.0

\[e^\pi - \pi \approx 19.9991\]

1760

in Marathi we often use “सतराशेसाठ” (1,760) to refer to a generically large quantity

https://twitter.com/ManishEarth/status/1470432828185878529

close to 2143

\[22 \pi^4 \approx 2143.0\]

3999

roman numerals

https://www.youtube.com/watch?v=jMxoGqsmk5Y&t=383s

65536 − 2^−37

~2^16

Handbook of Floating-Point Arithmetic

An even more striking behavior happens with some early versions of Excel 2007: When you try to compute 65536 − 2^−37 the displayed result is 100001. This is an error in the binary-to-decimal conversion used for displaying that result: the internal binary value is correct, if you add 1 to that result you get 65537.

An explanation can be found at https://www.microsoft.com/en-us/microsoft-365/blog/2007/09/25/calculation-issue-update/

1000000

~2^20

Mathematica switches formatting of reals

In[15]:= 999999.+1
Out[15]= 1.*10^6

16777216

2^24

single precision

16777216

16777216.0 + 1.0 == 16777216.0

2^40

2^40 as low-brow approximation of \omega

9007199254740992.0

9007199254740992 == 2^53

double precision

9007199254740992.0 + 1.0 == 9007199254740992.0

2^53 - 1 is maximum safe integer

21474836480413647819643794

~2^84

Handbook of Floating-Point Arithmetic

With the previous release (6.0) of the same system, when entering 21474836480413647819643794 you would get 413647819643790) +′ − − .(− − .(