The decimal number 2.074 can be converted into a fraction to provide a more precise and simplified representation. To achieve this, we start by expressing 2.074 as a fraction with the decimal part over 1, considering the place value of the last digit. Since 2.074 has three decimal places, we can express it as 2074/1000.
Conversion Process
The conversion of 2.074 into a simplified fraction involves finding the greatest common divisor (GCD) of 2074 and 1000 to reduce the fraction to its simplest form. The GCD is the largest number that divides both 2074 and 1000 without leaving a remainder.
Greatest Common Divisor (GCD) Calculation
To calculate the GCD of 2074 and 1000, we can use the Euclidean algorithm or factorization. The Euclidean algorithm involves a series of division steps, where we divide the larger number by the smaller one and then replace the larger number with the remainder of the division, repeating the process until the remainder is 0. The last non-zero remainder is the GCD.
| Step | Division | Quotient | Remainder |
|---|---|---|---|
| 1 | 2074 ÷ 1000 | 2 | 74 |
| 2 | 1000 ÷ 74 | 13 | 42 |
| 3 | 74 ÷ 42 | 1 | 32 |
| 4 | 42 ÷ 32 | 1 | 10 |
| 5 | 32 ÷ 10 | 3 | 2 |
| 6 | 10 ÷ 2 | 5 | 0 |
From the table, we see that the last non-zero remainder before the division results in a remainder of 0 is 2, which is the GCD of 2074 and 1000.
Simplification of the Fraction
Now that we know the GCD of 2074 and 1000 is 2, we can simplify the fraction 2074⁄1000 by dividing both the numerator and the denominator by 2. This gives us 1037⁄500.
Actual Representation and Usage
The fraction 1037⁄500 represents the decimal 2.074 in a simplified and exact form. This representation is crucial in various mathematical, scientific, and engineering applications where precision is key. Understanding how to convert decimals to fractions and simplify them is a fundamental skill that enhances the accuracy and reliability of calculations and expressions.
Technical Specifications and Applications
In technical and scientific contexts, the ability to convert between decimal and fractional representations of numbers is vital. For instance, in electronics, the value of components like resistors and capacitors is often expressed in fractional form for precision. Similarly, in chemistry, the proportions of substances in a mixture are critical and are best represented in fractional terms for accuracy.
The conversion of 2.074 to the simplified fraction 1037/500 demonstrates the importance of understanding and applying mathematical principles to achieve precision and clarity in various fields of study and application.
What is the simplified fraction of the decimal 2.074?
+The simplified fraction of the decimal 2.074 is 1037⁄500, obtained by converting 2.074 into a fraction and then simplifying it by dividing both the numerator and denominator by their greatest common divisor, which is 2.
Why is it important to simplify fractions?
+Simplifying fractions is important because it provides a clearer and more precise representation of the value. It enhances the accuracy and reliability of mathematical expressions and calculations, especially in scientific, engineering, and technical applications where precision is critical.
