Constructibility of a decimal number

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In summary, the conversation discusses the concept of constructible numbers and how to determine if a given number, in this case 3.146891, is constructible. The person asking the question mentions using a straightedge and compass to perform constructions and mentions the theorem that states if a positive constructible number is given, then its square root is also constructible. They also bring up the rational root theorem and how it applies to terminating decimals. In the end, it is concluded that 3.146891 is indeed constructible since it can be represented as a rational number.
  • #1
trap101
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This a question I have in a number therory course. I've been asked to determine if

3.146891 is constructible.

Since there are probably many ways to solve this I should give a flavor of what I know and maybe then it would be easier to determine the level of detail needed.

So from what I know: a real number is constructible if the point corresponding to it on the number line can be obtained from the marked points 0 and 1 by performing a finite sequence of constructions using only a straightedge and compass.

Another theorem that I feel might be of importance is the fact that if "r" is a positive constructible number, then √r is constructible.

In other examples I used the rational root theroem, but I wasn't dealing with decimals. I just found reading elsewhere that all terminating decimals have a rational representation of the form:

K/ 2n5m...this was never covered in class, but if this is the case that would mean that 3.146891 is constructible because it is a rational number and I've shown that the rational numbers are consturctible.

Seems long winded in my opinion. Is this the right rationale?
 
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  • #2
trap101 said:
This a question I have in a number therory course. I've been asked to determine if

3.146891 is constructible.

Since there are probably many ways to solve this I should give a flavor of what I know and maybe then it would be easier to determine the level of detail needed.

So from what I know: a real number is constructible if the point corresponding to it on the number line can be obtained from the marked points 0 and 1 by performing a finite sequence of constructions using only a straightedge and compass.

Another theorem that I feel might be of importance is the fact that if "r" is a positive constructible number, then √r is constructible.

In other examples I used the rational root theroem, but I wasn't dealing with decimals. I just found reading elsewhere that all terminating decimals have a rational representation of the form:

K/ 2n5m...this was never covered in class, but if this is the case that would mean that 3.146891 is constructible because it is a rational number and I've shown that the rational numbers are consturctible.

Seems long winded in my opinion. Is this the right rationale?

Of course, it's rational. It's 3146891/1000000.
 
  • #3
hmmmmm. I guess I made a big fuss over nothing then. Thanks.
 

1. What is meant by the constructibility of a decimal number?

The constructibility of a decimal number refers to the ability to represent the number using a finite sequence of mathematical operations starting from the number 0. This includes addition, subtraction, multiplication, division, and taking roots.

2. How can one determine if a decimal number is constructible?

A decimal number is considered constructible if it can be expressed using a finite number of steps involving basic mathematical operations starting from the number 0. This can be determined by examining the decimal expansion of the number and checking if it follows a repeating pattern or if it can be written in terms of known constructible numbers.

3. What is the significance of determining the constructibility of a decimal number?

Determining the constructibility of a decimal number is important in understanding the limitations of mathematical operations and their ability to represent numbers. It also has applications in geometry and algebraic field extensions.

4. Can all decimal numbers be constructed?

No, not all decimal numbers can be constructed. In fact, the majority of decimal numbers are not constructible. For example, numbers such as pi and e cannot be constructed as they have infinite and non-repeating decimal expansions.

5. Are there any alternative methods for representing non-constructible decimal numbers?

Yes, there are alternative methods for representing non-constructible decimal numbers, such as using continued fractions or expressing them in terms of other mathematical constants. However, these methods may not always provide an exact representation of the number and may only be approximations.

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