Question

Write the repeating decimal as a fraction in simplest form. \(0.444 \ldots\)

Short answer

The repeating decimal \(0.444\ldots\) can be written as the fraction \(4/9\) in simplest form.

Step-by-step solution

Set an equation

Let's denote the repeating decimal by \(x\). Thus, we have the equation \(x = 0.444... \)

Multiply by 10

To get rid of the decimal point, multiply \(x\) (and its corresponding right side) by 10. This gives us another equation, \(10x = 4.444...\)

Subtract the first equation from the second one

This subtraction will help to get rid of the repeating part. By subtracting Eq.1 from Eq.2, we get \(10x - x = 4 \), which simplifies to \(9x = 4 \)

Solve for x

To solve the equation for \(x\), divide both sides of the equation by 9. This gives \(x = 4/9\)

Key concepts

Repeating Decimals

Repeating decimals are numbers that have one or more repeating numbers after the decimal point. For example, in the decimal \(0.444\ldots\), the digit 4 repeats indefinitely. This means it goes on forever with 4s appearing in the same sequence or cycle without end. Understanding repeating decimals is crucial because they show how some decimal numbers never come to an exact end, making them special and interesting in math.

• A repeating decimal can be identified if it has infinite digits with one or more digits repeating.
• These repeating sequences can often represent fractions, which allows them to be expressed in a more concise form.
• Tools like bar notations \(\overline{4}\) can be used to indicate the repeating part of a decimal, helping simplify the writing of such numbers.

Recognizing how to convert repeating decimals is a key concept in making them more manageable in algebraic expressions.

Fractions

Fractions are mathematical expressions representing the division of one whole number by another. In essence, they show us parts of a whole. So how do repeating decimals connect to fractions? When you have a repeating decimal, it often corresponds exactly to a rational fraction, which has a simple relationship between its numerator and denominator.

• A repeating decimal can always be expressed as a fraction where either the numerator or the denominator reflects its repeating nature.
• For instance, the decimal \(0.444\ldots\) equals the fraction \(4/9\) in its simplest form.
• The conversion involves using algebraic steps to link the infinite decimal form to a simple fraction form.

Learning how repeating decimals relate directly to fractions allows us to use them in arithmetic calculations more effectively and make sense of their otherwise endless nature.

Mathematical Equations

Mathematical equations are statements that assert the equality of two expressions. They are fundamental tools used to solve problems in mathematics and to express relationships between different quantities. In the context of converting repeating decimals to fractions, equations are used to establish and solve relationships that link decimals with fractions.

• To convert a repeating decimal to a fraction, we first set up an equation to represent the decimal. For example, \(x = 0.444\ldots\).
• By manipulating this equation, such as multiplying both sides by 10, we shift the decimal and align the repeating parts.
• Through subtraction of the two equations set and solving for \(x\), the repeating decimal is transformed into an algebraic fraction, such as \(x = 4/9\).

These algebraic steps are vital, as they allow for a clear conversion from a repeating infinite sequence to a precise fractional representation, shedding light on the inherently repeating nature of these decimals.