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In my text book, it is written that $\ce{NH3}$ is a base. They mentioned the reason for this is that it produces $\ce{OH-}$ ions in water. They stated this reaction:

$\ce{NH3 + H2O -> NH4OH}$

So That's why, they said that it is a base.

But in a neutralization reaction, mentioned by our teacher,

Acid + Base = Salt + water

So, if $\ce{NH3}$ is a base then, it should be,

Acid + $\ce{NH3}$ = Salt + Water

But it does not happen. When reacting with acid, it doesn't produce water.

For example,

$\ce{NH3 + HCl -> NH4Cl}$

-- Reference

Again,

$\ce{NH3 + H2SO4 -> (NH4)2SO4}$

-- Reference

So it is not producing water in the neutralization reaction. Hence, I'm confused whether it is base or not.

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Your problems are caused by using the Arrhenius theory, which is based on electrolytic dissociation.

According to the Arrhenius definition, acids are compounds that dissociate and release hydrogen ions $(\ce{H+})$ into the solution: $$\ce{HCl -> H+ + Cl-}$$ Bases are defined as compounds that dissociate and release hydroxide ions $(\ce{OH-})$ into the solution: $$\ce{NaOH -> Na+ + OH-}$$ The products of a neutralization reaction of an acid with a base are a salt and water: $$\ce{HCl + NaOH -> NaCl + H2O}$$ This definition works for metal hydroxides such as $\ce{NaOH}$, which can exist as pure compounds. However, it doesn’t work for bases such as $\ce{NH3}$.

In order to solve this problem for $\ce{NH3}$, the fictitious compound “ammonium hydroxide” $\ce{NH4OH}$ was proposed (see also this related answer). Assuming this formula, the Arrhenius definitions of bases and neutralization reactions work again: $$\ce{NH4OH -> NH4+ + OH-}$$ $$\ce{HCl + NH4OH -> NH4Cl + H2O}$$ However, ammonium hydroxide cannot be isolated; the fictitious solid compound does not exist. Nevertheless, the misleading traditional name “ammonium hydroxide” is still widely used for solutions of ammonia in water.

In order to really solve your problems with $\ce{NH3}$, you need to switch from the definition of bases according to the Arrhenius theory to the definition of bases according to the Brønsted–Lowry theory.

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Well, you can explain it as a lewis base. In ammonia, nitrogen forms 3 single covalent bonds with 3 atoms of hydrogen. However it has 5 electrons in its valence shell. Which means that there are two electrons left a.k.a lone pair of electrons. Hence, it acts as a lone pair electron doner making it a lewis base.

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"Acid + base gives salt and water" is only possible when acid and base both can dissociate into two ions. Let's take an example
$\ce{HCl}$ and $\ce{NaOH}$ gives $\ce{NaCl}$ and $\ce{H2O}$
So, in that case, $\ce{HCl}$ and $\ce{NaOH}$ both can dissociate.

But in case of $\ce{NH3}$ and $\ce{H2O}$ only $\ce{H2O}$ dissociate into $\ce{H+}$ and $\ce{OH-}$ but $\ce{NH3}$ remains the same.
And reaction proceeds with the help of lone pair of $\ce{NH3}$.
$$\ce{NH3 + H+ + OH- ->NH4+ +OH-}$$ and in later stages both combine together due to electrostatic force.

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