Fact Family – Truth Mafia or Math Term?

Have you heard of a fact family? Does it make you think of the math mob? Or the truth police? Here's what it is, really. — Have you heard of a fact family? Does it make you think of the math mob? Or the truth police? Here’s what it is, really.

“What the heck is a fact family?” I asked the first time I read the term.

And the second, third, fourth and fifth times.

“Fact families” – both the term and the families – seem to be everywhere these days.

So what IS a fact family?

Well, it’s a family of facts. But not parents or cousins. More like siblings.

Here’s a nice one I’ve known for years:

$8 = 3 + 5$
$5 = 8 - 3$
$3 = 8 - 5$

And here’s another one I’m fond of:

$42 = 6 \times 7$
$6 = 42 \div 7$
$7 = 42 \div 6$

Is that a real thing in math?

The short answer is “no.”

When you look it up on WolframAlpha.com, you don’t get a definition. It gives you the definition of “family.”

And according to that definition, any collection of facts (totally random, even) can be a “fact family.”

So is it a real thing?

The long answer is “yes.”

A nice little collection of related facts is called a fact family.

But sometimes we get even sloppier and say that 8, 3 and 5 are a fact family. The numbers aren’t the facts, but, as grownups, we know what we mean.

I’ve mentioned a number of times that mathematicians just make this crap up. So even WE can play a mathematician and create…

A Formal Definition bum-bum-buum

Ahem… pardon me while I geek out a moment…

Let $(G,\bigstar)$ be a group and $a$ , $b$ and $c$ in $G$ so that $a \bigstar b=c$ . Then we say $a$ , $b$ and $c$ form a fact family and we can write

$a\bigstar b=c$ $b=-a\bigstar c$ $a = c\bigstar -b$

A Casual Definition

We can call any three numbers $a$ , $b$ and $c$ , that make $a+b=c$ , a fact family. And we know that not only does $a+b=c$ , but also $c-b=a$ and $c-a=b$ .

Likewise with multiplication, we can call any three numbers $a$ , $b$ and $c$ , that make $a\times b=c$ , a fact family. And we know that not only does $a \times b=c$ , but also $c \div b=a$ and $c \div a=b$ .