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I've heard that $\ce{NAD^+}$ gains a Hydrogen proton during glycolysis and the Krebs cycle and becomes reduced to $\ce{NADH}$. However, isn't reduction when a molecule receives an electron?

Maybe I've been misinformed? Does $\ce{NADH}$ gain a Hydrogen electron, not a proton?

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You are correct that reduction is simply a gain of electrons. This results in a decrease in oxidation number.

You know that NAD+ is reduced by this process because it starts off with a positive charge (+1) and ends up with a neutral charge (0).

The reducing agent that is donating the electrons is the hydrogen. More correctly, the electrons come from the hydride (H-).

The hydride is represented by 2 electrons on this redox diagram: enter image description here

As you can see, the reduction reaction transfers a proton (H+) and hydride electrons (H- or 2e-) to NAD+. One electron goes to the nitrogen, and the other to the carbon where the proton binds. More details on the redox reaction of NAD can be found here.

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    $\begingroup$ Hydride anions do not exist in biochemical reactions. $\endgroup$ – Chris Dec 5 '17 at 19:53

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