Decimal Expansion

Definition

A decimal expansion is the representation of a number in the decimal number system, where the number is expressed as a sequence of digits to the right of a decimal point. This representation allows for the expression of fractional and irrational numbers with infinite decimal places.

5 Must Know Facts For Your Next Test

1. Decimal expansions can be used to represent both rational and irrational numbers, with rational numbers having either terminating or repeating decimal expansions.
2. The decimal expansion of a fraction can be found by dividing the numerator by the denominator, with the resulting digits forming the decimal expansion.
3. Irrational numbers, such as $\pi$ and $\sqrt{2}$, have infinite, non-repeating decimal expansions, which means their digits never form a repeating pattern.
4. The decimal expansion of a number can be used to approximate the value of that number to any desired degree of accuracy by considering a finite number of decimal places.
5. Decimal expansions are crucial in various mathematical and scientific applications, including measurement, calculations, and the representation of continuous quantities.

Review Questions

• Explain how the decimal expansion of a fraction can be found, and provide an example.
  • The decimal expansion of a fraction can be found by dividing the numerator by the denominator. For example, to find the decimal expansion of the fraction $\frac{1}{3}$, we divide 1 by 3, which results in the decimal expansion 0.333... (a repeating decimal). This is because the division process never terminates, as 3 does not divide evenly into 1, and the remainder of 1 is repeatedly divided by 3, leading to the infinite repetition of the digit 3.
• Describe the characteristics of irrational numbers and their decimal expansions.
  • Irrational numbers, such as $\pi$ and $\sqrt{2}$, are numbers that cannot be expressed as a simple fraction. Their decimal expansions are infinite and non-repeating, meaning that their digits never form a repeating pattern. This is because irrational numbers are not the solution to any equation with integer coefficients. As a result, their decimal expansions are unpredictable and cannot be fully determined, making them fundamentally different from rational numbers, which have either terminating or repeating decimal expansions.
• Explain how decimal expansions can be used to approximate the value of a number, and discuss the importance of this in various applications.
  • Decimal expansions can be used to approximate the value of a number to any desired degree of accuracy by considering a finite number of decimal places. This is particularly useful in various mathematical and scientific applications, such as measurement, calculations, and the representation of continuous quantities. For example, in engineering and physics, decimal expansions are used to express measurements with the appropriate level of precision, while in computer programming, decimal expansions are used to represent real numbers with a finite number of bits. The ability to approximate numbers using decimal expansions is crucial for making accurate calculations, measurements, and decisions in these and many other domains.

Related terms

Terminating Decimal: A decimal expansion that ends after a finite number of decimal places, such as 0.375 or 2.5.

Repeating Decimal: A decimal expansion where a digit or a sequence of digits repeats infinitely, such as 0.333... or 0.142857142857...

Irrational Number: A number that cannot be expressed as a simple fraction, and therefore has an infinite, non-repeating decimal expansion, such as $\pi$ or $\sqrt{2}$.