Frational base & 2019

Algebra Level 3

What is the approximate representation of $0.2019$ in base $2.019$ ?

0.0111 0101 1001 0.0011 0111 1000 0.0011 0101 1000 0.0011 1111 1000 0.1011 1101 1000

1 solution

David Vreken
Dec 29, 2018

To convert the decimal fraction $0.2019$ to base $2.019$ , first multiply the decimal and base together, and if the product is less than $1$ then write a $0$ and transfer the product to the next line, but if the product is more than $1$ then write a $1$ and transfer the product minus $1$ to the next line. Then repeat the process of multiplying the decimal and base and carrying the new number to the next line. The numbers in the last column will be the numbers after the decimal place for the number in the new base.

So in this case:

0.201900 \times 2.019 = 0.407636 \to 0

0.407636 \times 2.019 = 0.823017 \to 0

0.823017 \times 2.019 = 1.661672 \to 1

0.661672 \times 2.019 = 1.335916 \to 1

0.335916 \times 2.019 = 0.678214 \to 0

0.678214 \times 2.019 = 1.369313 \to 1

0.369313 \times 2.019 = 0.745643 \to 0

0.745643 \times 2.019 = 1.505454 \to 1

0.505454 \times 2.019 = 1.020511 \to 1

0.020511 \times 2.019 = 0.041411 \to 0

0.041411 \times 2.019 = 0.083609 \to 0

0.083609 \times 2.019 = 0.168807 \to 0

which means $0.2019_{10} \approx \boxed{0.001101011000_{2.019}}$ .

Frational base & 2019

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