We all know there’s a useful rule to follow when we think we have solved a problem in maths: does the answer look roughly right? Does it have a ball-park common-sense robustness about it?

If it looks screwy — I owe how much to HMRC? — it’s often worth searching for a misplaced zero.

But what if the sums and your feeling of what is ‘roughly right’ coincide perfectly because you are not only a dimwit when it comes to calculation but a you have no idea about wider truths either: you cannot do a ballpark calculation sensibly because you don’t know what a ballpark looks like?

This horrifying coincidence of mathematical and spatial myopias did great damage last week to the reputation of one of America’s best-know broadcasters and to MSNBC, the Left-wing Fox News equivalent that plays an important role in geeing up Democrats for the presidential fight.

I will get to why it matters so much in a moment but first the fun of the broadcast. It was a live interview between square-jawed Brian Williams (in 2007, one of Time magazine’s Top 100 most influential people in the world, though a bit of a lion in winter these days) and a woman called Mara Gay, who is a member of the New York Times editorial board, a former Wall St Journal reporter and a graduate of the University of Michigan at Ann Arbor, one of the top academic institutions in the United States.

The hypocrisy of billionaire-bashing By Justin Webb

They were doing what people do on cable news: discussing a tweet.

The tweet had said that Michael Bloomberg had spent $500 million on his presidential campaign. That bit was true. Then, noting that there are 327 million Americans — also true — the tweeter suggested that Bloomberg could have given each American a million dollars “and still had lunch money left over”.

Brian Williams went full Ron Burgundy.

“When I read it tonight on social media, it kind of all became clear,” he said.

“It’s an incredible way of putting it,”

“It’s an incredible way of putting it,” agreed Ms Gay. She added: “It’s true. It’s disturbing.”

Cue massive hilarity and embarrassment and piling-on. Numerate America pointed out that 500 million divided by 327 million is roughly 1.5. Not a million. Just one dollar and fifty cents could have been the gift Bloomberg made to everyone in the nation.

The hypocrisy of billionaire-bashing By Tom Chivers

Now, wait a moment. Highly educated people are often innumerate — UnHerd’s Tom Chivers frequently, if gently, points this out. It’s awful but it’s true. Effortlessly numerate commentators stand out because there are so few of them. One of the reasons why Evan Davis is such a wonderful asset to the BBC is that he can count: he can see numbers.

So let’s give Brian Williams and Mara Gay a pass on the maths. Let’s assume we are all of us (except Tom) in the same glasshouse here and we are eyeing the stones but deciding not to throw. No: let us concentrate instead on the context, the ballpark.

This is the real horror story. These two people, whose entire jobs can be described as knowing things about society, seriously believed (“It all became clear.” “True.” “Disturbing.”) that it might be possible for Michael Bloomberg to give every American a million dollars without it making any impact on his wealth. Every American. Folk in small-town diners. College kids. Donald Trump. My own daughter Clara (as a citizen, I assume she’d qualify — yippee!)

How could that feel right? How could it feel true? Why would he not have done it already and asked if they might consider voting for him when they cashed the cheque?

The answer is that for a certain group of Americans who want to see change but not pay for it, the idea has taken hold that all that needs to happen for everyone in the nation to be wealthy is for a few billionaires to give away part of their wealth: note ‘part of,’ not even much of it.

The hypocrisy of billionaire-bashing By Tanya Gold

The idea has skewed the debate about healthcare that seems likely to be the ground zero of the presidential campaign in the autumn.

Plenty of Sanders supporters seem to genuinely believe that American healthcare could become free at the point of delivery — entirely government funded — if a few wealthy people paid more tax. Perhaps they think Mike Bloomberg could fund it.

It’s barmy. And to be fair to Senator Sanders, he makes no such claims. Yes, he points out that according to the think tank the Institute for Policy Studies three people — Bill Gates, Warren Buffett and Jeff Bezos — own more wealth than the bottom half of the American population.

True. Disturbing, even.

But Sanders does not pretend that going after them and them alone is not enough to change America.

As well as “getting rid of billionaires” (he has a car bumper sticker promising this so it must be true) he acknowledges the need for the much less wealthy to chip in: employees would pay a 4% premium on all income above $29,000 to fund Medicare for All, he says. So once you earn serious money you would be paying a decent whack.

And here’s the rub. Williams seems to think — along with so many on the American Left — that it is not about him. The billionaires will foot the bill for fairness. But it’s nonsense. The brilliant English think-tanker Richard Reeves pointed out years ago — in his book Dream Hoarders — that the top 20% of the USA should be where the finger is pointed and the redistribution might begin. These, Reeves says, are the comfortably off who have used their wealth and their cultural power to cream off the best of American life for themselves.

The hypocrisy of billionaire-bashing By ISABEL SAWHILL AND CHRISTOPHER PULLIAM

“It would be an exaggeration to say that the upper-middle-class is full of gluten-avoiding, normal-BMI joggers who are only marginally more likely to smoke a cigarette than to hit their children,” Reeves wrote. “But it would be just that — an exaggeration, not a fiction.”

This is where the debate about American inequality should perhaps be focused: where it could make a real difference. It’s not (just) about the billionaires; it’s about the glass floor that separates the upper-middle-class of America from the rest.

It’s about persuading the gluten-free joggers to have respect for people who live in communities in which gender roles are still marked. In which family ties still matter more than any other. In which you run in the street only to get away from a mugger or the police.

Many of these folks rather like billionaires. They admire their energy. They envy them their yachts. Suggesting to them that life could be dandy if the billionaires coughed up is insulting to their intelligence, and avoids, according to those who study these matters seriously, the real problem.

In sleek studios in New York, it is easier to discuss tweets about billionaires and mangle maths than to focus on what would drive real change. That would require a talk about class. Where do millions of people like Brian Williams spend their holidays? Where do their kids go to university? And are they willing to give up their privilege? As a fellow broadcaster I suspect he’d employ one of our favourite get-out clauses — always true on the Today programme of course — “I’m so sorry, we’re out of time…”