Decimals

To Do: A restaurant offers a 10% discount. Do you prefer getting the discount before or after the 8% sales tax is added on?

First the discount, then the tax: P 0.9 1.08

First the tax, then the discount: P 1.08 0.9

Same

Think about how you might respond to a student who asks you the following question?

Why, when we add decimals, we have to line up the decimal points, while, when we multiply decimals, we don’t have to line up the decimal points?

Adding and

Multiplying Decimals

To Do: Add 0.3 + 0.4 by first expressing each as a fraction, then adding the fractions, and finally convert back to a decimal.

How does this compare with the way decimal addition usually is done?

To Do: Add 0.6 + 0.9 by first expressing each as a fraction.

How does this compare with the way decimal addition usually is done?

To Do: Add 0.3 + 0.57 by first expressing each as a fraction.

How does this compare with the way decimal addition usually is done?

To Do: Multiply 0.7×0.43 by first expressing each as a fraction.

How does this compare with the way decimal multiplication usually is done?

To Do: Suppose m

n is in lowest terms. Under

what circumstances does m

n have a terminating

decimal representation?

Answer: If the only prime divisors of n are 2 and/or 5.

Terminating and

Repeating Decimals

To Do: Suppose m

n is in lowest terms. Under

what circumstances does m

n have a repeating

decimal representation?

Answer: If there is a prime divisor of n other than 2 and 5.

To Do: How do we know that 1

7 has a

repeating decimal representation rather than a non-repeating decimal representation?

When 10, 20, 30, 40, 50 or 60 is divided by 7, the remainder can only be one of the numbers 1, 2, 3, 4, 5 or 6, which means we’ll again be dividing into either 10, 20, 30, 40, 50 or 60.

If m

n is a proper fraction for which n is not divisible by 2 or

5, then the period of its decimal expansion is how many 9’s are in the smallest number of the form 9999 for which n is a divisor.

To Do: What is the smallest number of the form 9999 for which 7 is a divisor?

7 i s n o t a d i v i s o r o f 9 o r 9 9 o r 9 9 9 o r 9 9 9 9 o r 9 9 9 9 9 . H o w e v e r , 7 i s a d i v i s o r o f 9 9 9 9 9 9 s i n c e

9 9 9 9 9 9 = 7 1 4 2 8 5 7 . S o , 1

7 h a s p e r i o d 6 . I n d e e d ,

1= 0 . 1 4 2 8 5 7 1 4 2 8 5 7 1 4 2 8 5 7

7

To Do: What is the smallest number of the form 9999 for which 13 is a divisor?

Indeed, 5=0.384615

13 (period =6).

To Do: What is the smallest number of the form 9999 for which 37 is a divisor?

Indeed, 12=0.324

37

1 5= 0 .8 8 2 3 5 2 9 4 1 1 7 6 4 7 0 5

1 7

If m

n is a proper fraction for which n is not divisible by 2 or

5, then the period of its decimal expansion is how many 9’s are in the smallest number of the form 9999 for which n is a divisor.

1

n has period 5 for what n?

99999 = 32 41 271

10.02439

41

10.00369

271

Convert 0.29 to a fraction

Let =0.29x

Then 100 =29.29x

So 99x = 29

29

99x

To Do: Convert 0.756 to a fraction

Let = 0.756x

Then 1000 = 756.756x

So 999x = 756

756 28

999 37x

If m

n is a proper fraction for which n is not divisible by 2 or

5, then the period of its decimal expansion is how many 9’s are in the smallest number of the form 9999 for which n is a divisor.

Why does this rule work?

Suppose we didn’t know the period of 1

7.

Write 1

=07

abcdefghijkl.

Then, 1000000

=7

abcdef.ghijkl

But, 10000001 999999

- =7 7 7

= a whole number

10000001 999999- =

7 7 7 = a whole number

Thus, their decimal parts must be equal. That is, g = a, h = b, i = c, j = d, k = e, and l = f. Therefore, the decimal repeats every 6 decimal places.

1= 0

7abcdefghijkl.

1000000=

7abcdef.ghijkl