Question

Prove that: (a) \(0.999999 \ldots 99 \ldots=1\) (b) \(0.34627 \overline{0}=0.34626 \overline{9}\).

Short answer

\(0.999\ldots = 1\); \(0.34627\overline{0} = 0.34626\overline{9}\).

Step-by-step solution

Understand the Decimal Representation 0.999...

The number 0.999... is a repeating decimal that goes on indefinitely with the digit 9. We need to show how this equals 1.

Set the Decimal as a Variable

Let \( x = 0.999\ldots \). This allows us to manipulate the number algebraically.

Multiply the Variable by 10

Multiply both sides of the equation by 10, giving \( 10x = 9.999\ldots \).

Subtract the Original Equation from the Multiplied Equation

Subtract the original \( x = 0.999\ldots \) from \( 10x = 9.999\ldots \), resulting in \( 9x = 9 \).

Solve for x

Divide both sides by 9 to get \( x = 1 \), proving \( 0.999\ldots = 1 \).

Understand the Decimal Transitions for (b)

The number \(0.34627\overline{0}\) suggests a repeating zero, whereas \(0.34626\overline{9}\) has repeating nines at the end. We aim to prove these are equivalent.

Convert Repeating Decimals to Fractions

For \(0.34627\overline{0}\), the repeating zero indicates it is the same as a fraction with the denominator 1 followed by as many zeros as there are repeating digits.

Recognize Decimal Equivalence

Convert \(0.34626\overline{9}\) by understanding that a repeating nine in the decimal shifts the final digit up by 1 over an infinitely small distance, hence \(0.34627\overline{0}\).

Conclude Equivalency for Part (b)

It follows that since both methods target the limit approaching a specific number, the decimals are equivalent: \(0.34627\overline{0} = 0.34626\overline{9}\).

Key concepts

Almost Equal Decimals

When we talk about almost equal decimals, we are referring to numbers like 0.999... and 1. At first glance, these might seem different, but in mathematical terms, they are actually the same. This is because 0.999... is a repeating decimal that approaches 1 infinitely close without ever actually differing from it.
The idea might be a bit tricky, but here's a simpler way to think about it:

• Imagine you're filling a cup with water to the top. The 0.999... is like continuously adding a bit more every time, getting ever closer to filling it completely. At infinity, it's indistinguishable from full.
• Both numbers possess the same value in the real number system because there is no real number between them.

Once you understand this, you see how 0.999... converges to the number 1 itself. This helps establish that they are almost equal and, in mathematical terms, indeed the same.

Infinite Series

Infinite series play a crucial role in understanding repeating decimals like 0.999... When we express repeating decimals in the form of series, we use sequences that go on indefinitely, known as infinite series. The decimal 0.999... can be represented as an infinite series:\[0.9 + 0.09 + 0.009 + ext{and so on}.\]This series can be understood as a geometric series where the sum converges to a specific limit, which is 1 in this case. Here's a breakdown:

• The formula for the sum of an infinite geometric series \(S = \frac{a}{1 - r}\), where \(a\) is the first term, and \(r\) is the common ratio, applies.
• For 0.999..., \(a = 0.9\) and \(r = 0.1\), leading to \(S = \frac{0.9}{1-0.1} = 1\).

This approach clearly demonstrates how repeating decimals equate to whole numbers like 1 and helps simplify these infinite series into comprehensible, finite numbers.

Repeating Decimals

Repeating decimals occur when a digit or group of digits repeats itself indefinitely after the decimal point. These can be expressed as fractions or understood through infinite series. Both 0.999... and examples like 0.34627\(\overline{0}\) fit this pattern. Here’s why they’re important:- They provide a deeper understanding of fractions and division outcomes. For instance, 1/3 results in 0.333..., showcasing how fractions represent repeating decimals.- In some cases, repeating decimals can denote numbers that are almost but not quite an integer, as seen with 0.999...For decimals like 0.34626\(\overline{9}\), the behavior of 9s infinitely repeating doesn’t change the value but adjusts the perception of decimal placement. Therefore:

• An understanding of these repeating patterns can make converting them into fractional form or algebraic equalities easier.
• It emphasizes how infinitely small differences become indistinguishable over an infinite sequence.

Algebraic Manipulation

Algebraic manipulation plays a significant role in proving equivalency between repeating decimals and other numbers. Setting the repeating decimal as a variable allows us to use mathematical operations to unravel its true value. Here’s how it works for 0.999...:

• Assign the decimal a variable, such as \(x = 0.999...\).
• Multiply by 10 to shift the decimal: \(10x = 9.999...\).
• Subtract the original \(x\) from this to isolate digits: \(10x - x = 9.999... - 0.999...\).
• This results in \(9x = 9\), and solving gives \(x = 1\).

This method provides a solid proof of how repeating decimals can equal whole numbers. Such algebraic manipulation simplifies complex-looking decimals into understandable forms, making equivalence and transitions in decimal representations clear and conclusive.