Does every real number have a unique decimal expression?

Every real number can be represented by its decimal representation, which consists of a finite list of digits, a decimal point, and an infinite list of digits. (You can always adjoin an infinite number of zeroes for numbers such as 2.56.) One such number is √2. Such numbers are called irrational numbers.

Is an infinite decimal a real number?

First of all, if we have defined real numbers as infinite decimals, then the procedure just outlined really does unambiguously define a real number. In fact, it is the unique infinite decimal x such that, for every n, x(n)2 < 2 and (x(n)+10-n)2 > 2.

Do you think the decimal representation of a number is unique?

In general, there are real numbers that can be expressed either as a number ending in an infinite string of nines or an infinite string of zeros. That’s the only way the real representations are not unique!

What can be represented by finite or infinite decimals?

Numbers with decimal expansions that are infinite (i.e., non-terminating) and do not have a repeat block are called irrational numbers. Numbers with finite (i.e., terminating) decimal expansions, as well as those numbers that are infinite with repeat blocks, are called rational numbers.

What is decimal representation of real numbers?

Decimal Representation of Terminating Rational Number The terminating decimal expansion means that the decimal representation or expansion terminates after a certain number of digits. A rational number is terminating if it can be expressed in the form: p/(2n×5m).

Is a Nonterminating Nonrepeating decimal?

A non-terminating, non-repeating decimal is a decimal number that continues endlessly, with no group of digits repeating endlessly. Decimals of this type cannot be represented as fractions, and as a result are irrational numbers. Pi is a non-terminating, non-repeating decimal.

What is the decimal representation of 136 upon 1400?

Hence, 0.0971 is the correct decimal representation of .

What is the decimal expansion of irrational number?

The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational. S = 0.10110111011110…

Why are some decimals infinite?

Another way to write an infinite decimal with a repeating pattern is to draw a bar over the part that repeats. There are also infinite decimals without repeating patterns. These decimals represent the irrational numbers, and there’s no way to know all the digits of any such number.

Can a real number have more than one decimal expansion?

Each real number has either one or two decimal representations— never more, never less. We can divide all decimal expansions into three categories: If the decimal expansion of x terminates, you can get a second representation by decrementing the last digit and appending infinite 9s. For example, 0.128≡0.127999¯9.

Does every real number have an infinite decimal representation?

Yes, every real number has a unique infinite decimal representation, if we interpret “infinite” as “does not eventually become all 0’s” (except for the number 0 itself, which we agree to represent as 0.0000 … ).

Is it possible to represent every real number in R?

The answer is yes. The fact that Q is dense in R gives us the start line to reach every real number as a sequence of increasing long sequence of digits. ( 1, 1.4, 1.41, 1.414, ⋯ → 2 ) to compute even more digits we have lots of algorithms. if your question is “I think about x random real number, what’s the decimal representation?”

How do you represent numbers with two representations?

Numbers of the form n / 10 k where n and k are integers and n ≠ 0 have two representations each (e.g. 1.000… and 0.999… represent the same real number). Depending on your conventions, zero has either one or two representations: e.g. 0 and − 0 in the latter case.

What is the definition of a real number?

Now, to your question: another reasonable definition of a real number is a non-terminating decimal expansion (we say non-terminating just to clear up the ambiguity that arises between e.g. 123.4599999… and 123.46 — only the first is allowed). It turns out that this definition is equivalent to all the others.