One third in decimal is represented as 0.333…, a repeating decimal. This is obtained by performing division between the numerator (1) and denominator (3) of the fraction 1/3. The decimal representation of 1/3 does not terminate but repeats the digit 3 indefinitely, representing the infinite nature of the fraction. Understanding the concept of decimals, fractions, and the mathematical operation of division is crucial for comprehending this conversion process.
Decimal: A Base-10 Number System
Decimals are a way of writing numbers using a base-10 system. This means that each digit represents a power of 10. The rightmost digit represents 10 to the power of 0, the next digit represents 10 to the power of 1, and so on.
For example, the number 345.67 can be broken down as follows:
- 3 * 10^2 = 3 * 100 = 300
- 4 * 10^1 = 4 * 10 = 40
- 5 * 10^0 = 5 * 1 = 5
- 6 * 10^-1 = 6 * (1/10) = 0.6
- 7 * 10^-2 = 7 * (1/100) = 0.07
Adding these up, we get 300 + 40 + 5 + 0.6 + 0.07 = 345.67.
The digits 0-9 are used to represent the different values in decimals. The digit 0 represents the absence of a value, so 0.07 means seven hundredths (7/100). The digit 1 represents the value of 10^0, or 1. The digit 2 represents the value of 10^1, or 10, and so on.
Fraction: A Part of a Whole
- Define fractions as a mathematical representation of a part or portion of a whole.
- Explain the components of a fraction: numerator (indicates the number of parts taken) and denominator (indicates the total number of parts in the whole).
Fraction: A Part of the Whole
Imagine dividing a delicious pizza into equal slices to share with friends. Fractions are like the slices, describing parts of a whole.
A fraction has two important components: the numerator and the denominator. The numerator tells us how many slices we have, while the denominator tells us how many slices the whole pizza is divided into.
For example, the fraction 1/3 means we have one slice of pizza, and the whole pizza is divided into three equal slices. It’s like having a third of the pizza.
Key Points:
- Fractions represent parts or portions of a whole.
- The numerator indicates the number of parts taken.
- The denominator indicates the total number of parts in the whole.
- Fractions allow us to describe and compare different parts of a whole.
One Third: A Fraction of the Whole
Imagine you have a delicious pizza, and you’re in the mood to share the joy. You decide to divide it equally among three friends. Each friend gets a slice that represents one-third of the whole pizza. This fraction is denoted as 1/3.
In mathematical terms, a fraction is a way to represent a part of a whole. It consists of two numbers: a numerator (1) and a denominator (3). The numerator tells us how many parts we have, while the denominator tells us the total number of parts in the whole.
So, 1/3 represents one out of three equal parts of a whole. For instance, if you have a pizza with 12 slices, each slice would represent 1/3 of the whole pizza.
To better visualize this, imagine a rectangular pizza cut into three equal columns. Each column represents one-third of the whole pizza. Now, take one of the columns – that’s your 1/3 portion.
Converting Fractions to Decimals: The Art of Division
In the realm of numbers, fractions and decimals dance in intricate harmony. A fraction embodies a part of a whole, like a slice of pie, while a decimal extends this concept to an infinite canvas of numbers after the decimal point. To bridge these two worlds, we embark on a journey of conversion, guided by the power of division.
Let’s consider a humble fraction: 1/3. This fraction represents one part of a whole that is divided into three equal parts. To transform this fraction into its decimal equivalent, we summon the trusty tool of long division.
Step by step, we embark on our division expedition:
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Divide and conquer: We place 1 (the numerator) inside the division symbol and 3 (the denominator) outside to the left.
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Perform the division: We divide 1 by 3, which yields a quotient of 0 and a remainder of 1.
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Bring down the decimal: We drop the decimal point to the quotient and bring down the next digit, which is 0.
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Continue dividing: We divide 10 by 3, which gives us a quotient of 3 and a remainder of 1.
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Repeat the process: We bring down the next digit, which is 0 again, and divide 100 by 3. This time, the quotient is 33, with no remainder.
Amazingly, our division adventure reveals that 1/3 expressed as a decimal is 0.333…. The ellipsis (…) signifies that the pattern of 3s will repeat infinitely, creating a non-terminating decimal.
This repeating decimal holds a profound secret: it mirrors the infinite nature of the fraction itself. No matter how many times we divide 1 by 3, we will never reach an exact answer. Thus, the decimal representation captures the essence of the fraction’s endless divisibility.
In essence, the conversion of a fraction to a decimal through division serves as a bridge between two distinct mathematical worlds. This process empowers us to express fractions in a form that is more familiar and versatile, while also shedding light on the underlying relationships between numbers.
Comprehending Repeating Decimals and the Concept of Infinity
In the realm of mathematics, not all numbers are as straightforward as they seem. Some numbers, like the decimal representation of fractions, have a quirky trait: they never end. These special numbers are known as repeating decimals. One prime example of a repeating decimal is the decimal equivalent of one-third: 0.333….
When we divide one by three, we embark on a mathematical journey that never truly concludes. The quotient, represented as a decimal, goes on and on forever, with the digit “3” repeating itself ad infinitum. This infinite repetition signifies that the fraction one-third is an irrational number, meaning it cannot be expressed as a simple fraction with whole numbers.
The concept of infinity, symbolized by the ∞ sign, plays a pivotal role in understanding repeating decimals. Just as a circle has no true beginning or end, the decimal representation of one-third has no discernible boundary beyond which it terminates. The repeating sequence of “3”s becomes a symbol of this infinite nature.
The fascinating behavior of repeating decimals opens up a whole new dimension in the realm of numbers. They challenge our intuitive understanding of what constitutes a number and force us to embrace the concept of infinity.
Decimal: A Base-10 Number System
Decimals are a number system based on powers of 10. We use the digits 0-9 to represent different values in decimals. For example, the digit 3 in the decimal number 345 represents the value 3 × 10^2 = 300.
Fraction: A Part of a Whole
Fractions are a mathematical representation of a part or portion of a whole. Fractions are written as two numbers separated by a line, called a vinculum. The top number, called the numerator, indicates the number of parts taken, while the bottom number, called the denominator, indicates the total number of parts in the whole.
One Third as a Fraction: 1/3
The fraction 1/3 represents a part of a whole divided into three equal parts. The numerator, 1, indicates that one part is taken, while the denominator, 3, indicates that the whole is divided into three equal parts.
Converting Fraction to Decimal: Division
We can convert a fraction to a decimal by performing division. To convert the fraction 1/3 to a decimal, we divide 1 by 3:
1 ÷ 3 = 0.333...
The decimal representation of 1/3 is 0.333…, where the digits 3 repeat infinitely.
Repeating Decimals and Infinity
Some fractions, like 1/3, cannot be expressed as a terminating decimal. These fractions are represented by repeating decimals, where the decimal representation does not terminate but repeats an infinite sequence of digits.
Mathematical Operation of Division
Division plays a crucial role in the conversion of fractions to decimals. The mathematical operation of division is the process of finding how many times one number (the divisor) is contained within another number (the dividend). In the conversion of fractions to decimals, we use division to express the relationship between the numerator and denominator.
For example, when we convert the fraction 1/3 to a decimal, we are essentially finding how many times 1 is contained within 3. The result, 0.333…, represents the infinite number of times that 1 can be divided into 3.
The decimal equivalent of one third is 0.333…, a repeating decimal. This conversion highlights the mathematical operation of division and the importance of understanding the relationship between the numerator and denominator in fractions.
