Algorithms for Operations with Decimals:

Addition and Subtraction:

Line up the decimal points and proceed as with whole number addition and

subtraction-carries and borrows. You might have to include missing zeros.

Examples:

Multiplication:

Example:

This is the same as . So multiply the whole numbers 43 and 17,

and then move the decimal point to the left three places. , so the answer

is .731.

Let’s try .

Division:

Example:

This is the same as . So we’ll use the

standard algorithm, keeping track of the decimal point:

Let’s try .

A reduced fraction whose denominator’s prime factorization has no primes other

than 2’s or 5’s has a terminating decimal representation.

Examples:

Find the decimal representation of .

A reduced fraction whose denominator’s prime factorization contains prime factors

other than 2 or 5 has a non-terminating, repeating decimal representation.

Examples:

So . The smallest length string of repeating digits is called the

repetend, and the number of digits in the repetend is called the period of the

repeating decimal. The horizontal overbar is called a vinculum.

So the decimal representation of has a repetend of 3 and therefore, a period of 1.

. What’s the repetend and period of the decimal representation

of ?

Let’s find the repetend and period of .

Given a non-terminating, repeating decimal, you can find a fraction that it’s

equivalent to.

Examples:

Let , then . If you subtract the first equation from the second

equation, you get . So .

Let , then . If you subtract the first equation from the second

equation, you get . So .

Find a fraction with the decimal representation .

Let , then .

Find a fraction with the decimal representation .

Let , then .