Some tourists in the Museum of Natural History are marveling at some dinosaur bones. One of them asks the guard, “Can you tell me how old the dinosaur bones are?”
The guard replies, “They are 3 million, four years, and six months old.”
“That’s an awfully exact number,” says the tourist. “How do you know their age so precisely?”
The guard answers, “Well, the dinosaur bones were three million years old when I started working here, and that was four and a half years ago.”
Lets be honest about what we know and what we do not know
We write the values that have real meaning
3 million years means 3±0.5 ⨉ 106
Adding 4.5 years is meaningless
Notice that 1234.0 is not the same as 1234
Also, 12340. is not the same as 12340
This is a convention but not everybody uses it
It is much safer to use scientific notation
It is safer to represent numbers in scientific notation \[ \begin{align} 1234.0 & = 1.2340\cdot 10^3\\ 1234 & = 1.234 \cdot 10^3\\ 12340. & = 1.2340 \cdot 10^4\\ 12340 & = 1.234 \cdot 10^4 \end{align} \]
All digits in the mantissa are significant
(mantissa is the number being multiplied by 10 to the power of the exponent)