I want to know if there is any difference between Fractions and Decimal numbers, are Decimal numbers just Fractions that are written in a different way according to a predefined rule: using "a group of Fractions" that each have a denominator that is smaller 10 times than the one before it.
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2$\begingroup$ It depends on what definitions you are using for those two terms. Sometimes, people use "fraction" to mean any real number between zero and one; sometimes, people use "fraction" to mean something of the form $a/b$ with $a$ and $b$ integers, $b$ positive. Gilles and I use different definitions of "decimal number". What definitions do you have in mind? $\endgroup$– Gerry MyersonCommented Feb 3, 2015 at 12:02
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$\begingroup$ I say, WHAT DEFINITIONS DO YOU HAVE IN MIND? $\endgroup$– Gerry MyersonCommented Feb 4, 2015 at 12:12
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$\begingroup$ Decimals are simply fractions where the divisor is some power of 10. For example $\frac{1}{4}=\frac{25}{100}$ usually written a $0.25$ and common knowledge lets us know that it is $25$ divided by $100$. If you type a decimal into a cell in Excel and then go to format cells, number and then select fraction 2-digit, you will see, for example. that $0.351\approx \frac{33}{94}$ $\endgroup$– poetasisCommented Oct 13, 2020 at 4:28
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[edited after reading the comments]
I think I learned as a child that a decimal number is a real number that can be written in the form $a/10^b$ where $a\in\mathbb{Z}$ and $b\in\mathbb{N}$. With this convention there exist rational numbers which are not decimal, e.g $1/3=0.3333...$
But after seeing the comments and having a look at wikipedia it seems this is not the most common definition. So if we consider that
A decimal number is any real number. The term decimal refers only that we are representing it in base ten.
A rational number is a real number that can be written in the form $a/b$ where $a,b\in\mathbb{Z}$ and $b\neq0$.
Then any rational number is a decimal number. But there exist decimal number which are not rational, e.g $\sqrt{2}=1.414...$.
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3$\begingroup$ Maybe we are using different definitions of "decimal number", but I would say $0.333\dots$ is a decimal number; I'd also say $\sqrt2=1.41421356\dots$ is a decimal number that can't be written as a fraction. $\endgroup$ Commented Feb 3, 2015 at 11:59
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$\begingroup$ @GerryMyerson Now we've reached a point where we need to agree on a definition of decimal numbers to continue, specifically whether our definition allows infinite sequences of digits. $\endgroup$ Commented Feb 3, 2015 at 12:42