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The collection of all rational and irrational decimal numbers is called the real numbers. Yes, there are “unreal” numbers too (called imaginary numbers), but we’re not going to go there! Any rational number can either be expressed as a. terminating decimal (i.e., a decimal with a finite number of nonzero digits such as 0.4)
Note that all decimals that terminate are rational numbers. In simple words, all terminating decimals are rational numbers. Thus, they can be written in the form of a fraction, as they are rational numbers. Example 1: 1.5 can also be written as $\frac{15}{10}$. Example 2: $\frac{1}{8} = 0.125$
Terminating decimal numbers has a finite number of digits after the decimal point. A number with a terminating decimal is always a rational number. If the denominator of a rational number can be expressed in form 2 p 5 q or 2 p or 5 q, where p,q∈N, then the decimal expansion of the rational number terminates.
Therefore every rational number is represented by a decimal that either terminates or repeats. Formal proof attempt: Claim: if a number is rational, then it's decimal expansion either terminates or repeats. Proof: let a/b be a rational number. Then by the definition of rational numbers a/b is a ratio of the integers a and b where b divides a.
Wikipedia claims that every repeating decimal represents a rational number. ... Every positive rational number has either a terminating or repeating decimal expansion ...
Jun 5, 2023 · However, not all decimal representations are rational numbers. A number written in decimal form where there is a last decimal digit (after a given decimal digit, all following decimal digits are 0) is a terminating decimal, as in 1.34 above. Alternately, any decimal numeral that, after a finite number of decimal digits, has digits equal to 0 ...
Sep 5, 2011 · I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly appreciated.
