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Rounding replaces a number by a permitted representable value according to a specified rule. Rounding to nearest minimizes absolute error; directed rounding instead selects the next permitted value toward positive infinity, negative infinity or zero. A tie between equally near values needs an explicit convention.
On the decimal grid with spacing , rounds to nearest as , with error . Rounding to nearest has error at most half the grid spacing. In normal floating-point arithmetic, that spacing depends on magnitude, leading to a relative-error bound described by unit roundoff. The IEEE round-to-nearest-even rule resolves ties by an even final significand digit.[ref]
IEEE arithmetic in Oracle's Numerical Computation Guide. Chapter 2, IEEE Arithmetic Model, four rounding directions
Fixed-point arithmetic represents values as integer multiples of a fixed scale . Integer operations on , followed by the required rescaling and rounding, implement arithmetic on these values. Unlike floating-point arithmetic, the spacing is constant throughout the representable range.
For , the stored integer represents . Multiplying two encoded values and gives , so returning to the original scale requires dividing the integer product by with a stated rounding rule. Fixed spacing makes resolution predictable, but the chosen integer width limits range and can cause integer overflow.
Fixed-point iteration seeks a fixed point of a map by repeatedly setting . Convergence requires additional hypotheses; for example, a contraction mapping on a complete metric space has a unique fixed point reached by this iteration.
The surreal numbers form an ordered field built recursively from cuts , where every already constructed left option is smaller than every right option. The earliest examples are , , and . A cut names the simplest number between its options; different cuts can name the same number.
Allowing construction through arbitrary ordinals introduces both infinite numbers, such as , and positive infinitesimals, such as . All real numbers embed in the resulting ordered field, but its entire collection is a proper class. Unlike ordinal arithmetic, surreal multiplication and addition obey the usual commutative field laws.
UDP-glucose is an activated donor containing a glucose residue linked to UDP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
UDP-galactose is an activated donor containing a galactose residue linked to UDP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
UDP-N-acetylglucosamine is an activated donor containing a N-acetylglucosamine residue linked to UDP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
GDP-mannose is an activated donor containing a mannose residue linked to GDP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
GDP-fucose is an activated donor containing a fucose residue linked to GDP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
ADP-glucose is an activated donor containing a glucose residue linked to ADP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
CMP-sialic acid is an activated donor containing a sialic acid residue linked to CMP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
UDP-glucuronic acid is an activated donor containing a glucuronic acid residue linked to UDP. Transfer requires an appropriate glycosyltransferase and acceptor; access to the correct compartment can be limiting.
Functionally coupled receptors whose sensor and signaling activities may be partitioned between two genes. Physical linkage can help preserve their compatible inheritance.
A domain incorporated into an immune receptor that mimics an effector target and links its perturbation to immune recognition.
An SN2 reaction couples backside nucleophilic bond formation to leaving-group departure in one elementary substitution step. At a suitable stereogenic reaction centre, the geometry produces inversion.
In an addition–elimination pathway, a nucleophile attacks an aromatic carbon and forms a Meisenheimer complex; leaving-group loss restores aromaticity. Electron-withdrawing groups ortho or para to the attacked site can stabilize the addition intermediate.
A limiting unimolecular nucleophilic substitution mechanism with rate-determining ionization, commonly producing a carbocation, followed by nucleophile capture. Its ideal rate law is first order in substrate.
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 2. You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website.
Figure 3. Visual Studio Code extension installation.
Figure 5. . You can also edit articles on the Web editor without installing anything locally. Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension. 
- Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact