Notation and Terminology :
A real number has a decimal representation. It gives the approximate location of the number on the number line.
Examples:
The rational number 1/2 is real and has the decimal representation 0.5. The rational number has the representation . The number 1/3 is also real and has the infinite decimal representation 1.333… This means there is an infinite number of 3’s, or to put it another way, for every positive integer n, the nth decimal place of the decimal representation of 1/3 is 3.
The number has a decimal representation beginning 3.14159… So you can locate approximately by going 3.14 units to the right from 0. You can locate it more exactly by going 3.14159 units to the right, if you can measure that accurately. The decimal representation of is infinitely long so you can theoretically represent it with as much accuracy as you wish. In practice, of course, it would take longer than the age of the universe to find the first 10 to the power of 10 to the power of 10 digits.
Bar notation:
It is customary to put a bar over a sequence of digits at the end of a decimal representation to indicate that the sequence is repeated forever. For example,
42 (1/3) = 42, 3bar.
and 52.71656565… (65 repeating infinitely often) may be written 52.7165 . 65 for 65 the bar notation is assign. means we have to put bar.
A decimal representation that is only finitely long, for example 5.477, could also be written 5.4770. in this 5.4770 for zero we have to put bar on that means we have to write bar(-) on zero.
Terminology:
The decimal representation of a real number is also called its decimal expansion. A representation can be given to other bases besides 10; more about that here.
Variations in usage:
Approximations:
If you give the first few decimal places of a real number, you are giving an approximation to it. Mathematicians on the one hand and scientists and engineers on the other tend to treat expressions such as " 3.14159" in two different ways.
The mathematician may think of it as a precisely given number, namely 314159 / 100000, so in particular it represents a rational number. This number is not pie, although it is close to it.
The scientist or engineer will probably treat it as the known part of the decimal representation of a real number. From their point of view, one knows 3.14159 to six significant figures.
Abstractmath.org always takes the mathematician's point of view. If I refer to 3.14159, I mean the rational number 314159 / 100000. I may also refer to pie as “approximately 3.15159…”.
Decimal representation and infinite series:
The decimal representation of a real number is shorthand for a particular infinite series (MW, Wik). Let the part before the decimal place be the integer n and the part after the decimal place be
α1,α2,α3........
where αi is the digit in the i th place. (For example, for ∏, n=3 , α1=1,α2=4,α3=1,and so forth.) Then the
the decimal notation n.α1,α2,α3.... . represents the limit of the series
The decimal representations of two different real numbers must be different. However, two different decimal representations can, in certain circumstances, represent the same real number. This happens when the decimal representation ends in an infinite sequence of 9’s or an infinite sequence of 0’s.
Example
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These equations are exact. <!--[if !ie]--><!--[endif]-->
is exactly the same number as 3.5. (Indeed, <!--[if !ie]--><!--[endif]-->
, 3.5, 35/10 and 7/2 are all different representations of the same number.)
Two proofs that <!--[if !ie]--><!--[endif]-->
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The fact that <!--[if !ie]--><!--[endif]-->
is notorious because many students simply don’t believe it is true. I will give two proofs here.
.......................................
A real number has a decimal representation. It gives the approximate location of the number on the number line.
Examples:
The rational number 1/2 is real and has the decimal representation 0.5. The rational number has the representation . The number 1/3 is also real and has the infinite decimal representation 1.333… This means there is an infinite number of 3’s, or to put it another way, for every positive integer n, the nth decimal place of the decimal representation of 1/3 is 3.
The number has a decimal representation beginning 3.14159… So you can locate approximately by going 3.14 units to the right from 0. You can locate it more exactly by going 3.14159 units to the right, if you can measure that accurately. The decimal representation of is infinitely long so you can theoretically represent it with as much accuracy as you wish. In practice, of course, it would take longer than the age of the universe to find the first 10 to the power of 10 to the power of 10 digits.
Bar notation:
It is customary to put a bar over a sequence of digits at the end of a decimal representation to indicate that the sequence is repeated forever. For example,
42 (1/3) = 42, 3bar.
and 52.71656565… (65 repeating infinitely often) may be written 52.7165 . 65 for 65 the bar notation is assign. means we have to put bar.
A decimal representation that is only finitely long, for example 5.477, could also be written 5.4770. in this 5.4770 for zero we have to put bar on that means we have to write bar(-) on zero.
Terminology:
The decimal representation of a real number is also called its decimal expansion. A representation can be given to other bases besides 10; more about that here.
Variations in usage:
Approximations:
If you give the first few decimal places of a real number, you are giving an approximation to it. Mathematicians on the one hand and scientists and engineers on the other tend to treat expressions such as " 3.14159" in two different ways.
The mathematician may think of it as a precisely given number, namely 314159 / 100000, so in particular it represents a rational number. This number is not pie, although it is close to it.
The scientist or engineer will probably treat it as the known part of the decimal representation of a real number. From their point of view, one knows 3.14159 to six significant figures.
Abstractmath.org always takes the mathematician's point of view. If I refer to 3.14159, I mean the rational number 314159 / 100000. I may also refer to pie as “approximately 3.15159…”.
Decimal representation and infinite series:
The decimal representation of a real number is shorthand for a particular infinite series (MW, Wik). Let the part before the decimal place be the integer n and the part after the decimal place be
α1,α2,α3........
where αi is the digit in the i th place. (For example, for ∏, n=3 , α1=1,α2=4,α3=1,and so forth.) Then the
the decimal notation n.α1,α2,α3.... . represents the limit of the series
The decimal representations of two different real numbers must be different. However, two different decimal representations can, in certain circumstances, represent the same real number. This happens when the decimal representation ends in an infinite sequence of 9’s or an infinite sequence of 0’s.
Example
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<!--[if !ie]-->
<!--[endif]-->
<!--[if !ie]--><!--[endif]-->
These equations are exact. <!--[if !ie]--><!--[endif]-->
is exactly the same number as 3.5. (Indeed, <!--[if !ie]--><!--[endif]--> , 3.5, 35/10 and 7/2 are all different representations of the same number.)Two proofs that <!--[if !ie]--><!--[endif]-->
<!--[if !ie]--><!--[endif]-->The fact that <!--[if !ie]--><!--[endif]-->
is notorious because many students simply don’t believe it is true. I will give two proofs here........................................
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