Understanding Long Division to Calculate 1/17 in Decimal Form Without Rounding Off

The process of calculating the decimal form of 1/17 using long division is a fascinating exercise that involves both arithmetic and number theory. This method allows us to understand the precise representation of the fraction in decimal form. By performing long division step-by-step, we can uncover the repeating decimal sequence and verify its correctness. This article will guide you through the process and highlight the mathematical principles behind it.

The Principle and Process of Long Division

When calculating 1/17 in decimal form, we start by setting up the long division problem. Essentially, we are dividing 1 (the numerator) by 17 (the denominator). The key here is to recognize that the division will result in a repeating decimal because the decimal expansion of 1/17 will repeat due to the properties of 17 and 10.

Multiplicative Property and Periodicity

One of the crucial properties of 17 is its relationship with powers of 10. Specifically, we use the fact that 1016 ≡ 1 (mod 17). This equivalence indicates that 1016 - 1 is divisible by 17. This periodicity in the powers of 10 helps us predict the length of the repeating cycle in the decimal expansion of 1/17.

Performing the Long Division

To begin the long division of 1 by 17, we place a decimal point in the quotient and add as many zeros as needed to the dividend (10000000000000000), ensuring that we can continue the process step-by-step:

Step 1: Initial Setup

1 / 17

0. 1 17 100

Divide 100 by 17, which gives 5 with a remainder of 15.

Step 2: Continuing the Division

5 8 17 150 150 8 17 160 94 85 75 75 82 31 26 5 10 7 30 28 20 17 30 28 20 17 30

At this stage, we start to see a repeating pattern: 0588235294117647. This periodic sequence confirms that the decimal expansion of 1/17 is indeed a repeating decimal.

Verifying the Repeating Decimal

To verify the correctness of the repeating decimal sequence, we can use the relation:

x 0.0588235294117647

By multiplying both sides by 1016 (which is 10000000000000000), we get:

1016 x 588235294117647.0588235294117647

Then, subtract x from this equation:

1016 x - x 588235294117647 - 0.0588235294117647

Which simplifies to:

1016 - 1) x 588235294117647

Solving for x, we get:

x 1 / 17

This confirms that the repeating decimal 0.0588235294117647 truly represents the fraction 1/17 when the division is performed to the maximum possible precision without rounding off.

Conclusion

The process of calculating 1/17 in decimal form using long division is a perfect demonstration of the intersection between arithmetic and number theory. By understanding the properties of 17, specifically its behavior in modular arithmetic, we can predict and confirm the repeating decimal sequence. This method not only provides an accurate representation of the fraction but also underscores the elegance and complexity of mathematical patterns.

Related Keywords

long division decimal form repeating decimal