# Are 3.0, 3.00, 3.000 ... all equal?

This may be too trivial to ask. But when I searched, I didn't find anyone asking this question previously on this site.

When I consider 3.0 and 3.00, from a mathematical perspective, absolutely I can see both of them being equal. But some long time ago, I heard that those two are actually not equal in a scientific perspective, because they differ in number of significant digits after decimal point. Now, I wanted to know is this claim correct? Or just a hoax?

• they are all equal because $0$'s after the decimal point don't matter and that's because $0000\dots0 = 0$ Jul 22, 2021 at 5:13
• In a scientific or engineering context, the number of decimal places may indicate the accuracy of the measurement, so that $3.0$ means $3\pm.1$ and $3.00$ means $3\pm.01$ (or something of the sort) so they're not really the same. Similarly, if I say that something is one and a half meters long, it conveys something different than if I say it's $150$ centimeters long. Most people would take the second measurement to be more accurate, I think. Jul 22, 2021 at 5:13
• They are equal, because $$3.000...=3+\frac{0}{1000...}=3+0=3$$ Jul 22, 2021 at 5:32
• Draw a right triangle with $1$ on both sides. The length of the hypotenuse will vary depending on the precision of your gauge. $$1.4, 1.41, 1.414, 1.4142, ...$$ Mathematically the length is clear: $\sqrt 2.$ But in real life you'll never know.Maybe the length of what you are measuring is $3.000001$, but because the measuring instrument does not have enough precision, you will find the required length $3.000=3$. Jul 22, 2021 at 7:25
• @lonestudent quick question. Is $\sqrt 2$ equal to any number of the form 1.41... mathematically? All 1.41... to any precision is still an approximation of $\sqrt 2$. Am I right? Jul 22, 2021 at 7:54