Question

Does \(\frac{1}{3}\) equal 0.333 ? If not, which is larger? By how much?

Short answer

No, \(\frac{1}{3}\) is larger. The difference is 0.0003...

Step-by-step solution

Understand both numbers

The first number is a fraction: \(\frac{1}{3}\) and the second number is a decimal: 0.333.

Convert the fraction to a decimal

Divide 1 by 3 to convert \(\frac{1}{3}\) to a decimal. \(\frac{1}{3} = 0.3333...\). The decimal representation of \(\frac{1}{3}\) is a repeating decimal, 0.3333...

Compare the two decimals

Compare 0.333 to 0.3333... and determine which is larger. Since 0.3333... (repeating) continues infinitely, it is slightly larger than 0.333 which ends after three decimal places.

Calculate the difference

Subtract 0.333 from 0.3333... to find the difference. \[0.3333... - 0.333 = 0.0003...\].

Key concepts

Fractions

Fractions are a way to represent a part of a whole. They are written in the form \(\frac{a}{b}\), where \(a\) is the numerator (the top part) and \(b\) is the denominator (the bottom part).
The numerator indicates how many parts we have, while the denominator shows into how many equal parts the whole is divided.
For example, \(\frac{1}{3}\) means we have 1 part out of 3 equal parts of a whole.
Fractions can be converted into decimals to make them easier to compare with other numbers in decimal form.
To do this, we perform the division of the numerator by the denominator. For example, \(\frac{1}{3}\) converted to a decimal by dividing 1 by 3, which is approximately 0.3333...
However, it's important to remember not to round or truncate this value when we need an exact comparison.

Repeating Decimals

Repeating decimal, also known as a recurring decimal, is a decimal number that has digits which repeat infinitely.
For example, 0.3333... is a repeating decimal where the digit '3' goes on forever.
In many cases, repeating decimals are represented with a bar notation over the repeating digit(s). For example, \(0.\bar{3}\), where the bar over the 3 means that it repeats indefinitely.
Repeating decimals can sometimes make comparisons tricky because we need to recognize that they continue infinitely.
For instance, when comparing 0.333 (a decimal that stops) with 0.3333... (a repeating decimal), it's clear that 0.3333... is slightly larger because it has more 3s extending forever.

Decimal Representation

Decimals are numerical values that include a decimal point to show a fraction of a whole number.
Decimals make it easier to perform operations like addition, subtraction, multiplication, and division.
When converting fractions to decimals, the goal is to determine the decimal representation of a fraction.
For instance, dividing the numerator 1 by the denominator 3 in the fraction \(\frac{1}{3}\), we obtain the repeating decimal 0.3333... It's essential to write enough digits to accurately represent the repeating nature of the number.
Comparing decimals can be straightforward when the decimals have the same number of digits, but when one is repeating and the other is not, ensure to expand the repeating decimal adequately to make accurate comparison.
Understanding these nuances helps when solving problems like determining whether \(\frac{1}{3}\) equals 0.333 or not and by how much they differ.