Check out our list of the best sites to download free STL files, 3D printer files, 3D print models, and 3D printing designs in other file formats. For example, since 0.1 is not exactly 1/10, summing three values of 0.1 may not yield exactly 0.3, either:. 3 False Also, since the 0.1 cannot get any closer to the exact value of 1/10 and 0.3 cannot get any closer to the exact value of 3/10, then pre-rounding with round function cannot help.
Floating-point numbers are represented in computer hardware as base 2 (binary)fractions. For example, the decimal fraction
has value 1/10 + 2/100 + 5/1000, and in the same way the binary fraction
has value 0/2 + 0/4 + 1/8. These two fractions have identical values, the onlyreal difference being that the first is written in base 10 fractional notation,and the second in base 2.
Unfortunately, most decimal fractions cannot be represented exactly as binaryfractions. A consequence is that, in general, the decimal floating-pointnumbers you enter are only approximated by the binary floating-point numbersactually stored in the machine.
The problem is easier to understand at first in base 10. Consider the fraction1/3. You can approximate that as a base 10 fraction:
or, better,
or, better,
and so on. No matter how many digits you’re willing to write down, the resultwill never be exactly 1/3, but will be an increasingly better approximation of1/3.
In the same way, no matter how many base 2 digits you’re willing to use, thedecimal value 0.1 cannot be represented exactly as a base 2 fraction. In base2, 1/10 is the infinitely repeating fraction
Stop at any finite number of bits, and you get an approximation. On mostmachines today, floats are approximated using a binary fraction withthe numerator using the first 53 bits starting with the most significant bit andwith the denominator as a power of two. In the case of 1/10, the binary fractionis
3602879701896397/2**55
which is close to but not exactlyequal to the true value of 1/10.Many users are not aware of the approximation because of the way values aredisplayed. Python only prints a decimal approximation to the true decimalvalue of the binary approximation stored by the machine. On most machines, ifPython were to print the true decimal value of the binary approximation storedfor 0.1, it would have to display
![Keep Keep](https://www.ruoto.fi/storage/product_images/7/tuotesivu_FoxBullets.223Rem36g20pcs_3830067345427_fb0a272e04edc23ab2ec5ba7878495f2_1.jpg)
That is more digits than most people find useful, so Python keeps the numberof digits manageable by displaying a rounded value instead
Just remember, even though the printed result looks like the exact valueof 1/10, the actual stored value is the nearest representable binary fraction.
Interestingly, there are many different decimal numbers that share the samenearest approximate binary fraction. For example, the numbers
0.1
and0.10000000000000001
and0.1000000000000000055511151231257827021181583404541015625
are allapproximated by 3602879701896397/2**55
. Since all of these decimalvalues share the same approximation, any one of them could be displayedwhile still preserving the invariant eval(repr(x))x
.Historically, the Python prompt and built-in
repr()
function would choosethe one with 17 significant digits, 0.10000000000000001
. Starting withPython 3.1, Python (on most systems) is now able to choose the shortest ofthese and simply display 0.1
.Note that this is in the very nature of binary floating-point: this is not a bugin Python, and it is not a bug in your code either. You’ll see the same kind ofthing in all languages that support your hardware’s floating-point arithmetic(although some languages may not display the difference by default, or in alloutput modes).
For more pleasant output, you may wish to use string formatting to produce a limited number of significant digits:
It’s important to realize that this is, in a real sense, an illusion: you’resimply rounding the display of the true machine value.
One illusion may beget another. For example, since 0.1 is not exactly 1/10,summing three values of 0.1 may not yield exactly 0.3, either:
Also, since the 0.1 cannot get any closer to the exact value of 1/10 and0.3 cannot get any closer to the exact value of 3/10, then pre-rounding with
round()
function cannot help:Though the numbers cannot be made closer to their intended exact values,the
round()
function can be useful for post-rounding so that resultswith inexact values become comparable to one another:Binary floating-point arithmetic holds many surprises like this. The problemwith “0.1” is explained in precise detail below, in the “Representation Error”section. See The Perils of Floating Pointfor a more complete account of other common surprises.
As that says near the end, “there are no easy answers.” Still, don’t be undulywary of floating-point! The errors in Python float operations are inheritedfrom the floating-point hardware, and on most machines are on the order of nomore than 1 part in 2**53 per operation. That’s more than adequate for mosttasks, but you do need to keep in mind that it’s not decimal arithmetic andthat every float operation can suffer a new rounding error.
While pathological cases do exist, for most casual use of floating-pointarithmetic you’ll see the result you expect in the end if you simply round thedisplay of your final results to the number of decimal digits you expect.
str()
usually suffices, and for finer control see the str.format()
method’s format specifiers in Format String Syntax.For use cases which require exact decimal representation, try using the
decimal
module which implements decimal arithmetic suitable foraccounting applications and high-precision applications.Another form of exact arithmetic is supported by the
fractions
modulewhich implements arithmetic based on rational numbers (so the numbers like1/3 can be represented exactly).If you are a heavy user of floating point operations you should take a lookat the NumPy package and many other packages for mathematical andstatistical operations supplied by the SciPy project. See <https://scipy.org>.
Python provides tools that may help on those rare occasions when you reallydo want to know the exact value of a float. The
float.as_integer_ratio()
method expresses the value of a float as afraction:Since the ratio is exact, it can be used to losslessly recreate theoriginal value:
The
float.hex()
method expresses a float in hexadecimal (base16), again giving the exact value stored by your computer:This precise hexadecimal representation can be used to reconstructthe float value exactly:
Since the representation is exact, it is useful for reliably porting valuesacross different versions of Python (platform independence) and exchangingdata with other languages that support the same format (such as Java and C99).
Another helpful tool is the
math.fsum()
function which helps mitigateloss-of-precision during summation. It tracks “lost digits” as values areadded onto a running total. That can make a difference in overall accuracyso that the errors do not accumulate to the point where they affect thefinal total:15.1. Representation Error¶
This section explains the “0.1” example in detail, and shows how you can performan exact analysis of cases like this yourself. Basic familiarity with binaryfloating-point representation is assumed.
Representation error refers to the fact that some (most, actually)decimal fractions cannot be represented exactly as binary (base 2) fractions.This is the chief reason why Python (or Perl, C, C++, Java, Fortran, and manyothers) often won’t display the exact decimal number you expect.
Why is that? 1/10 is not exactly representable as a binary fraction. Almost allmachines today (November 2000) use IEEE-754 floating point arithmetic, andalmost all platforms map Python floats to IEEE-754 “double precision”. 754doubles contain 53 bits of precision, so on input the computer strives toconvert 0.1 to the closest fraction it can of the form J/2**N where J isan integer containing exactly 53 bits. Rewriting
as
and recalling that J has exactly 53 bits (is
>=2**52
but <2**53
),the best value for N is 56:That is, 56 is the only value for N that leaves J with exactly 53 bits. Thebest possible value for J is then that quotient rounded:
Since the remainder is more than half of 10, the best approximation is obtainedby rounding up:
Therefore the best possible approximation to 1/10 in 754 double precision is:
Dividing both the numerator and denominator by two reduces the fraction to:
Note that since we rounded up, this is actually a little bit larger than 1/10;if we had not rounded up, the quotient would have been a little bit smaller than1/10. But in no case can it be exactly 1/10!
So the computer never “sees” 1/10: what it sees is the exact fraction givenabove, the best 754 double approximation it can get:
If we multiply that fraction by 10**55, we can see the value out to55 decimal digits:
meaning that the exact number stored in the computer is equal tothe decimal value 0.1000000000000000055511151231257827021181583404541015625.Instead of displaying the full decimal value, many languages (includingolder versions of Python), round the result to 17 significant digits:
![Keep It 1 3 3 Keep It 1 3 3](https://images-na.ssl-images-amazon.com/images/I/61Qv5AZpONL.jpg)
The
fractions
and decimal
modules make these calculationseasy:It was Albert Einstein who said; “If you can’t explain it, you don’t understand it well enough.” Though it is often mis-reported as being; “If you can’t explain it to a six year old, you don’t understand it well enough.” What Einstein was driving at was a particular application of “keep it simple, stupid”.
From scientific concepts to products the end-user doesn’t care how clever the creator or designer of something is. They care about being able to take that person’s output and make it useful to their own lives. The simpler the explanation and the simpler the product, the more likely it is that the output will be useful to others.
The phrase; “keep it simple, stupid” is thought to have been coined by the late Kelly Johnson, who was the lead engineer at the Lockheed Skunk Works (a place responsible for the S-71 Blackbird spy plane amongst many other notable achievements). It is worth noting that Kelly’s version of the phrase had no comma and was written “keep it simple stupid”.
Author/Copyright holder: Terretta. Copyright terms and licence: CC BY-NC-ND 2.0
File list export 2 4 0 free. There’s really not much more to say here is there? Keep it simple stupid.
Kelly explained the idea to others with a simple story. He told the designers at Lockheed that whatever they made had to be something that could be repaired by a man in a field with some basic mechanic’s training and simple tools. The theater of war (for which Lockheed’s products were designed) would not allow for more than that. If their products weren’t simple and easy to understand – they would quickly become obsolete in combat conditions and thus worthless.
Today the KISS principle is celebrated in many engineering professions (including software engineering) and is often brought to bear by managers in many professions as well as by trainers and educators.
The First Usability Principle?
KISS may have been the first usability principle for product design – though it was never formally presented as a usability principle. It focuses on the idea that if we can’t understand a product, we can’t use it properly and that the widest possible audience must be able to understand it, if the product is to gain maximum market share. This is as true for mobile applications as it is for fighter planes.
Author/Copyright holder: United States Navy. Copyright terms and licence: Public Domain.
The Lockheed F-35 will have been built to the KISS principle and so should your products if you want them to succeed.
Variants of KISS
The KISS principle is also offered in two other forms (for those who feel delicate about the inclusion of the word “stupid”):
- Keep it short and simple
- Keep it simple and straightforward
Though both phrases technically introduce an “a” into the acronym – they both deliver the same message as “keep it simple, stupid”. The objective of any process is to deliver the simplest possible outcome.
Alternatives to KISS
KISS is related to a fair number of other famous quotes, phrases and principles. Among them:
- Occam’s Razor - “Among competing hypotheses, the one with the fewest assumptions should be selected” (but often restated as “The simplest solution is most likely the correct solution” which is not quite the same thing).
- Albert Einstein’s – “Make everything as simple as possible but not simpler” (it is possible that Einstein never actually said this and it was actually a paraphrase of something he said during a lecture but the principle remains sound).
- Leonardo Da Vinci’s – “Simplicity is the ultimate sophistication” (when, perhaps, the greatest designer in history offers this advice, it’s almost certainly good advice).
- Ludwig Mies Van Der Rohe’s – “Less is more” (Mies was a highly respected architect and peer of the better known Frank Lloyd Wright)
- Bjarne Stroustrup’s “Make Simple Tasks Simple!” (Stroustrup is a Danish computer scientist and highly regarded academic).
- Antoin Marie Jean-Baptiste Roger, comte de Saint-Exupery’s “It seems that perfection is not reached when there is nothing left to add but when there is nothing left to take away.”
Author/Copyright holder: Fred the Oyster. Copyright terms and licence: CC BY-SA 3.0
Occam’s razor is a metaphorical rather than literal razor and it’s often misquoted too. It is possible for something to be too simple and this diagram, rather ably, demonstrates.
1/3 Times 3 Equals
A Note of Caution When Applying KISS to Design
Whilst simplicity is an admirable goal and can lead to enhanced user experiences, it is important not to let simplicity interfere with the design objective. The user must still be able to carry out their task requirements with the finished products or the design process has failed – no matter how simple the final design.
Products such as DSLR Cameras are by nature more complex than the cameras found on the latest generations of smartphones. Complexity is to be resisted when it exists for its own sake and not when complexity can enhance the design for the user.
Author/Copyright holder: Bill Bertram. Copyright terms and licence: CC BY-SA 3.0
The DSLR is as simple as it can be without reducing its utility. It’s not as simple as a mobile phone camera but it offers more options to the photographer. KISS has not been abandoned here but rather kept in line with user expectations.
The Take Away
Simplicity is a key design principle. The easier something is to understand and use – the more likely it is to be adopted and engaged with. KISS, “keep it simple, stupid” is thus a great rule of thumb to be applied when considering your design work in a larger context of usage. However, it is also important not to make things so simple that they compromise the functionality of the final design – users will live with a little complexity if it enhances their overall experience.
1/3 Times 3
References
Forbes Magazine explores why using KISS doesn’t make you any stupider - http://www.forbes.com/sites/amyanderson/2014/02/27/keeping-it-simple-doesnt-mean-youre-stupid/#5ab32dc572ca
Lifehacker offers tips into implementing the KISS principle in your life in general here - http://www.lifehack.org/articles/communication/10-tips-kiss-your-life.html
Keep It 1 3 3 49
Hero Image: Author/Copyright holder: Kristian Bjornard. Copyright terms and licence: CC BY-SA 2.0