tzedek and tzedaka

In modern Hebrew, tzedek צדק and tzedakah צדקה have very different meanings. Tzedek is "justice, which is obligatory and compels all. Tzedaka is "charity", which is praiseworthy, but voluntary. (In Jewish law, giving charity in general is obligatory, but the amount given and the intended recipient is left to the donor's discretion.)

Both words are found in biblical Hebrew. Tzedek is found 119 times in the Bible, and tzedaka appears 157 times. In the Bible, they are essentially synonyms. They both refer to righteousness and justness. Nissan Netzer, in his book on Bereshit (p. 47), points out that there are synonym pairs in Biblical Hebrew where one word ends with the letter heh and the other doesn't. He brings the examples of otzem עוצם and otzma עצמה - which both mean "force, might", and shir שיר and shira שירה - which both mean "song."

This article by the Academy of the Hebrew Language points out that there is a slight difference between the two words in biblical Hebrew. Tzedek more often refers to the concept or value of justice, whereas tzedaka is more frequently found referring to the act (or acts) of performing justice. Evidence to this difference can be found by the fact that tzedek is only found in the singular, but tzedaka can have a plural (tzedakot צדקות).

Hebrew seems to have a hard time hanging on to synonyms. These differences in nuance, through a process known as "semantic shift", led the two words to diverge fully. Starting in Rabbinic Hebrew, they ended up as "justice" and "charity to the poor" (as an expression of justice). (The same phenomenon can be found with shir and shira. Today shir still means song, but shira refers to poetry.)

From the same root we get other Hebrew words. A tzadik  צדיק is a righteous person. And it also provides the verbs tzodek צודק -  "to be correct" and matzdik מצדיק - "to justify." These words seems to have echoes in other Semitic languages, as seen in the etymology Klein provides for the root:

Aram. צְדֵק (= he was righteous), Syr. זָדֵק (= it is right; for the change of צ to ז see the introductory article to letter ז), Ugar. ṣdq (= reliability, virtue), Arab. ṣadaqa (= he spoke the truth), Ethiop. ṣadaqa (= he was just, was righteous)

The connection between "correct" and "justice" can be found in English as well, in the related words "right" (correct) and "righteous."