Which statements are true for irrational numbers written in decimal form?
A. Irrational numbers are nonterminating.
B. Irrational numbers are repeating.
C. Irrational numbers are nonrepeating.
D. Irrational numbers are terminating.
A. Irrational numbers are nonterminating.
C. Irrational numbers are nonrepeating.
Well, irrational numbers tend to be indecisive when it comes to ending (nonterminating) and they also like to avoid repetition (nonrepeating). It's like they're saying, "Why settle for a simple pattern when you can have endless chaos?"
The correct statements about irrational numbers written in decimal form are:
A. Irrational numbers are nonterminating.
C. Irrational numbers are nonrepeating.
Irrational numbers are numbers that cannot be expressed as a fraction of two integers. When written in decimal form, they continue indefinitely without repeating a pattern of digits. Therefore, irrational numbers are nonterminating and nonrepeating.
The correct statements are A. Irrational numbers are nonterminating and C. Irrational numbers are nonrepeating.
To understand why these statements are true, we first need to understand the terms "terminating" and "repeating" in relation to decimal numbers.
1. Terminating decimals: These are decimals that have a finite number of digits after the decimal point. For example, 0.25 or 3.1416.
2. Repeating decimals: These are decimals that have a repeating pattern of digits after the decimal point. For example, 0.333... or 0.857857857...
Now, let's consider irrational numbers. An irrational number is a real number that cannot be written as a simple fraction (ratio) and has an infinite number of non-repeating decimal places. Examples of irrational numbers include √2, π, and e.
Statement A. Irrational numbers are nonterminating: This statement is true. Because irrational numbers have an infinite number of decimal places, they cannot be written as a fraction with a finite number of digits. Therefore, their decimal representation is nonterminating.
Statement B. Irrational numbers are repeating: This statement is false. Repeating decimals, as mentioned earlier, have a repeating pattern of digits. Irrational numbers, in contrast, have non-repeating decimal digits. They may appear random, but their digits do not follow a specific repeating pattern.
Statement C. Irrational numbers are nonrepeating: This statement is true. As mentioned earlier, irrational numbers have non-repeating decimal digits. There is no pattern to their decimal representation, making them nonrepeating.
Statement D. Irrational numbers are terminating: This statement is false. Terminating decimals, as mentioned earlier, have a finite number of digits. Irrational numbers have an infinite number of non-repeating decimal places, making them non-terminating.
In conclusion, the correct statements for irrational numbers written in decimal form are: A. Irrational numbers are nonterminating and C. Irrational numbers are nonrepeating.