From Webster's 1828 Dictionary: Righteousness, as used in Scripture and theology, in which it is chiefly used, is nearly equivalent to holiness, comprehending holy principles.
Is the Decalogue a standard of righteousness?
1. Do not make other gods.
2. Do not make idols.
3. Do not profane the name of YHWH.
4. Keep the Sabbath day holy.
5. Honor your parents.
6. Do not murder.
7. Do not commit adultery.
8. Do not steal.
9. Do not lie.
10. Do not covet.
Of the ten commandments, 8 are prohibitions against sin and 2 are commands to do something. The 4th command, as spoken by Moses in Exodus 20:8-11, provides instructions on how to keep it (rest from all work). The 5th command does not, merely giving a reason to obey.
Only God can know if a man worships idols or covets, though man may discern evidence of these sins.
How do these instructions establish righteousness?
Contrast the teachings of Moses found in the Decalogue with those of Paul in 1 Corinthians 13 and consider: which truly describes righteousness? Perfect love, described in 1 Cor 13, is an attribute of Christ Jesus, alone among men. And He is our righteousness.
Consider what Jesus revealed as the greatest commandments: Matthew 22:37-39 (HCSB) He said to him, “Love the Lord your God with all your heart, with all your soul, and with all your mind. This is the greatest and most important command. The second is like it: Love your neighbor as yourself." These two commands reflect a high standard of righteousness, one no mortal man can attain. Might these two words, which are NOT a summation of the Decalogue (see verse 40), be the highest standard given to us?