WASHINGTON — He offered to explain the intricacies of the federal budget. She reminded him that she had majored in math.

From the moment they knew they would be colleagues, representing the same state, it was clear the relationship would not be a great one.

Meet the Senate’s oddest couple: Ron Johnson, Republican, and Tammy Baldwin, Democrat, of Wisconsin, the personification of this polarized Congress, right down to their mahogany desks at opposite ends of the Senate chamber. No two senators from the same state vote against each other more often than they do — 75 percent of the time in 2013, a review of Senate records shows.

And when they do find patches of common ground it is often on uncontroversial nominees to lower courts and innocuous legislation like a bill to privatize the federal helium reserve. He, an uncompromising fiscal conservative, had never run for anything before the Tea Party wave of 2010 helped sweep him into office. She, a member of the House for 14 years who championed universal health care and stricter financial regulations, got her start in politics while still in law school.