the righteousness of God: This important phrase appears eight times in Romans (see also 3:5, 21, 22, 25, 26; 10:3; the only other occurrence in Paul's writings is 2 Corinthians 5:21). In the Old Testament, God’s righteousness refers to who God is (holy and faithful) and to what God does when he rescues his people and makes things right (see especially Isaiah 46:13; 51:5–8). Paul uses the second meaning in this verse. The gospel (the good news about Jesus) has the power to save because it fulfills God's promise to defend his people.
"The righteous will live by faith": The prophet Habakkuk struggled to understand how God could use nations that worshiped false gods to judge his own people, Israel. God reminded Habakkuk that his true people were the righteous, who needed to live by faith (Habakkuk 2:4). In Romans 1–4, Paul often insists that only by faith can humans become right with God.