Terminating decimals are decimals that end, meaning they have a finite number of digits after the decimal point. For example, 0.75 or 3.2 are terminating decimals. On the other hand, repeating decimals are decimals that go on indefinitely, with a pattern of digits that repeats infinitely. For example, 0.333... or 1.414141... are repeating decimals.
The end behavior of a decimal can help determine whether it is a rational number or an irrational number. Rational numbers are numbers that can be expressed as a fraction of two integers, while irrational numbers cannot be expressed as a simple fraction. Terminating decimals are always rational numbers, as they can be expressed as a fraction with a denominator that is a power of 10. Repeating decimals can also be rational, as they can be converted into a fraction using algebraic techniques. On the other hand, non-repeating decimals that do not terminate are typically irrational numbers, such as the square root of 2 or pi.
What is the difference between terminating and repeating decimals? How does the end behavior of a decimal relate to rational number or irrational number?
7 answers
Explain why a decimal for ⅓ is called a repeating decimal.
When we express the fraction 1/3 as a decimal, we get 0.333..., where the digit 3 repeats infinitely. This type of decimal is called a repeating decimal because it consists of a pattern of digits that repeats indefinitely.
To see why 1/3 results in a repeating decimal, we can consider the process of long division. When we divide 1 by 3, the quotient is 0 with a remainder of 1. We then bring down another zero and divide 10 by 3, resulting in a quotient of 3 with a remainder of 1. This process repeats indefinitely, always resulting in a quotient of 3 with a remainder of 1, leading to the repeating pattern of 0.333... after the decimal point.
Therefore, the decimal representation of 1/3 is a repeating decimal with the digit 3 recurring infinitely.
To see why 1/3 results in a repeating decimal, we can consider the process of long division. When we divide 1 by 3, the quotient is 0 with a remainder of 1. We then bring down another zero and divide 10 by 3, resulting in a quotient of 3 with a remainder of 1. This process repeats indefinitely, always resulting in a quotient of 3 with a remainder of 1, leading to the repeating pattern of 0.333... after the decimal point.
Therefore, the decimal representation of 1/3 is a repeating decimal with the digit 3 recurring infinitely.
Area Formula for a square and rectangle- define:
Quantitative relationships- define:
Interpreting Irrational answers for time and money- define:
Quantitative relationships- define:
Interpreting Irrational answers for time and money- define:
1. Area Formula for a square and rectangle:
- The formula to calculate the area of a square is side squared (A = s^2), where "s" represents the length of one side of the square. Since all sides of a square are equal, we only need to know the length of one side to calculate the area.
- For a rectangle, the area formula is length multiplied by width (A = l * w), where "l" represents the length of the rectangle and "w" represents the width. The area of a rectangle is the product of its length and width.
2. Quantitative relationships:
- Quantitative relationships refer to the manner in which quantities or numerical values are related to each other. In mathematics, these relationships describe how one quantity changes in relation to another, such as through addition, subtraction, multiplication, or division. Quantitative relationships can be expressed through mathematical equations, inequalities, graphs, or tables to demonstrate how quantities are connected and interact with each other.
3. Interpreting Irrational answers for time and money:
- When dealing with time or money, irrational answers may not be practical or realistic in certain contexts. For example, if we calculate the square root of a non-perfect square to determine the time it takes to complete a task or the cost of an item, the result may be an irrational number. In such cases, it is important to interpret these irrational answers in a meaningful way, such as rounding to a reasonable decimal or fraction that makes sense within the given scenario. This can help provide a more practical understanding of the time or cost involved, despite the irrational nature of the answer.
- The formula to calculate the area of a square is side squared (A = s^2), where "s" represents the length of one side of the square. Since all sides of a square are equal, we only need to know the length of one side to calculate the area.
- For a rectangle, the area formula is length multiplied by width (A = l * w), where "l" represents the length of the rectangle and "w" represents the width. The area of a rectangle is the product of its length and width.
2. Quantitative relationships:
- Quantitative relationships refer to the manner in which quantities or numerical values are related to each other. In mathematics, these relationships describe how one quantity changes in relation to another, such as through addition, subtraction, multiplication, or division. Quantitative relationships can be expressed through mathematical equations, inequalities, graphs, or tables to demonstrate how quantities are connected and interact with each other.
3. Interpreting Irrational answers for time and money:
- When dealing with time or money, irrational answers may not be practical or realistic in certain contexts. For example, if we calculate the square root of a non-perfect square to determine the time it takes to complete a task or the cost of an item, the result may be an irrational number. In such cases, it is important to interpret these irrational answers in a meaningful way, such as rounding to a reasonable decimal or fraction that makes sense within the given scenario. This can help provide a more practical understanding of the time or cost involved, despite the irrational nature of the answer.
Interpreting Irrational answers for time and money definition
Interpreting irrational answers for time and money involves making practical sense of the results obtained when dealing with calculations that involve time durations or monetary values. In real-world scenarios, irrational numbers might not be practical or meaningful when considering time intervals or financial amounts. To interpret irrational answers for time and money, we might round the result to a reasonable decimal or fraction that is more applicable to the specific context. This allows for a more understandable and practical representation of the time duration or monetary value, even though the initial calculation may have resulted in an irrational number. By rounding to a more conventional and sensible numerical format, we can better grasp the implications of the time or money involved in a given situation.