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1 Grow or cause to grow and become more mature, advanced, or elaborate.

no object
with object
grow , evolve, mature, expand, enlarge, spread, advance, progress, prosper, succeed, thrive, get on well, flourish, blossom, bloom, burgeon, make headway, be successful
initiate , instigate, set in motion, put in place, institute, inaugurate, originate, invent, form, establish, fashion, generate
expand , enlarge, add to, flesh out, supplement, reinforce, augment, extend, broaden, fill out, embellish, enhance, elaborate, amplify, refine, improve, polish, perfect

2 Start to exist, experience, or possess.

no object
with object
come into being , come about, start, begin, be born, come into existence, appear, arrive, come forth, emerge, erupt, burst out, arise, originate, break, unfold, crop up, follow, happen, result, ensue, break out
fall ill with , be taken ill with, be struck down with, be stricken with, succumb to

3 with object Treat (a photographic film) with chemicals to make a visible image.

4 Chess with object Bring (a piece) into play from its initial position on a player's back rank.


Mid 17th century (in the sense ‘unfold, unfurl’): from French développer, based on Latin dis- ‘un-’ + a second element of unknown origin found also in envelop.



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This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. [55] He proved the associative and commutative properties, among others, through mathematical induction .

Defining (−2) + 1 using only addition of positive numbers: (2 − 4) + (3 − 2) = 5 − 6.
Further information: Integer

The simplest conception of an integer is that it consists of an Bally VitaParcours sneakers outlet reliable free shipping classic cheap sale fashion Style cheap sneakernews YrsYiA
(which is a natural number) and a (generally either positive or outlet 2014 unisex Philippe Model trainers with glitter effect big sale cheap price 0ySRKnB
). The integer zero is a special third case, being neither positive nor negative. The corresponding definition of addition must proceed by cases:

Although this definition can be useful for concrete problems, it is far too complicated to produce elegant general proofs; there are too many cases to consider.

A much more convenient conception of the integers is the Grothendieck group construction. The essential observation is that every integer can be expressed (not uniquely) as the difference of two natural numbers, so we may as well define an integer as the difference of two natural numbers. Addition is then defined to be compatible with subtraction:

Addition of discount high quality adidas Outdoor Caprock Mens WaterResistant Hiking Shoes free shipping best D42qoFd29
can be computed using the least common denominator , but a conceptually simpler definition involves only integer addition and multiplication:

As an example, the sum 3 4 + 1 8 = 3 × 8 + 4 × 1 4 × 8 = 24 + 4 32 = 28 32 = 7 8 {\displaystyle {\frac {3}{4}}+{\frac {1}{8}}={\frac {3\times 8+4\times 1}{4\times 8}}={\frac {24+4}{32}}={\frac {28}{32}}={\frac {7}{8}}} .

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