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Density Property of Real Numbers

Real numbers are used to measure the continuous values. A real number can be rational number or irrational number or it may be positive, negative or zero. It may also be algebraic or transcendental.


According to the density property, the Real Numbers are infinite. Between two real numbers lies an infinite other real numbers. For examples between 3 and 4 lies $\frac{3}{2}$, $\frac{3}{4}$, $\frac{3}{5}$, $\frac{3}{7}$, $\frac{3}{8}$, $\frac{3}{11}$,....... etc. This means that real numbers cannot be counted and there can exist uncountable number of real numbers in the number system.
 

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To prove that the real numbers are dense:

  1. Consider any two real numbers say $\frac{3}{4}$ and $\frac{3}{8}$.
  2. Add them $\frac{3}{4}$ + $\frac{3}{8}$ = $\frac{9}{8}$
  3. Divide it by 2 and it gives the mid-value of $\frac{3}{4}$ and $\frac{3}{8}$. So, $\frac{9}{16}$ lies between $\frac{3}{4}$ and $\frac{3}{8}$.
  4. Similarly, we can find the mid-value of $\frac{3}{4}$ and $\frac{9}{16}$ and this will continue.

Hence it is proved that the real numbers are dense.

Find how many real numbers exist between 1 and 2. Can you count these numbers? Does it satisfy the density property of real numbers?

If we look at these two Numbers as Natural Numbers, we observe that there exist no number between two numbers, but it we take these numbers as real numbers, We find the number between 1 and 2 by = ( 1 + 2 ) /2, = 3/2. = 1.5, so we observe that 1.5 is a real number which exist between 1 and 2. Now in the same way we again check how many real numbers exist between 1 and 1.5. For this again we apply the same formula and get, = ( 1 + 1.5 ) /2, = 2.5 / 2 = 1.25. Thus 1.25 is a real number between 1 and 1.5, so it also lies between 1 and 2 also. In the same way we can proceed and see that this chain is endless. Thus we come to a conclusion that there are infinite real numbers between two real numbers. We also observe that these numbers cannot be counted, So we come to a conclusion that real numbers are dense, So it satisfies the Density Property of Real Numbers.

If we have two real numbers say 2 /5 and 2, does their exist any real number between them? If yes then how many real numbers can you find? Name the property so called?

We have the pair of real Numbers 2 / 5 and 2. To find the real numbers between these two numbers, we first add the two numbers and then divide them by 2 Sp we get, = ( 2 / 5 ) + 2, = ( 2 / 5 ) + ( 2 /1 ). Taking L.C.M of 5 and 1 as 5, we first make the denominator same For this, we multiply the numerator and the denominator of 2 by 5, we get, = ( 2 / 5 ) + 2*5/5 == ( 2 / 5 ) + ( 10 / 5 ) = ( 2 + 10 ) / 5 = 12 / 5 Now dividing the above number by 2, we get the real number which lies between = ( 2 / 5 ) and 2 = ( 12 / 5 ) ÷ 2 = (6 / 5 ) Now if we proceed in the same way we observe that there exist a real number between 2/5 and 6 / 5. Also there exist a real number between 6/5 and 2. This check can go up to an endless process and we will find a new number every time. Thus this list is endless and so there are infinite numbers between the given two numbers. This property of real numbers, where we can find infinite real numbers between any two real numbers is called the Density Property of Real Numbers.

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