Download Suppose you know that 7 flogs = 2 gloops where flogs and gloops

Suppose you know that 7 flogs = 2 gloops where flogs and
gloops are units of measure of something.
1 flog is 2/7 gloop and 1 gloop is 7/2 flog
10 miles per hour is how many kilometers per second
10 miles/hours = ? kilometers/second where we use
1
1
1
1
1
kilometer = 1000 meters
meter = 39 inches
foot = 12 inches
mile = 5280 feet
hour = 3600 seconds
10 miles/hour
= 10 miles/hour
= 10 miles/hour x 1 hour/3600 seconds
= 10 miles/3600 second
= 1/360 miles/second
= 1/360 miles/seconds x 5280 feet/miles
= 1/360 x 5280 feet/second
= 1/360 x 5280 feet/second x 12 inches/foot
= 1/360 x 5280 x 12 inches/second
= 1/30 x 5280 inches/second
= 1/30 x 5280 inches/second x 1 meter/39 inches
= 1/30 x 5280 x 1/39 meters/second
= 1/30 x 1/39 x 5280 meters/second x
1 kilometer/1000 meters
= 1/30 x 1/39 x 1/1000 x 5280 kilometers/second
= 1/1170000 x 5280 kilometers/second
= 5280/1170000 kilometers/second
= 0.00451282051282051... kilometers/second
= 0.00451[282051] kilometers/second
= 5.0 x 10-3 kilometers/second
Binary Numbers
The binary number system is 2 symbols – 0 and 1 – and is a
positional number system.
1112 =
=
=
=
1 x 102 + 1 x 101 + 1 x 100 in base 2
1 x 22 + 1 x 21 + 1 x 20 in base 10
4 + 2 + 1 in base 10
710
4710 = ?2
Using the division algorithm, we see that
47/2
23/2
11/2
5/2
2/2
1/2
=
=
=
=
=
=
2
2
2
2
2
2
x 23 + 1
x 11 + 1
x 5 + 1
x 2 + 1
x 1 + 0
x 0 + 1
4710 = 1011112
Adding binary numbers
11101
+01110
101011
11101 = 1 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 16 + 8 + 4 + 1
= 29
01110 = 0 x 24 + 1 x 23 + 1 x 22 + 1 x 21 + 0 x 20
= 0 + 8 + 4 + 2 + 0
= 14
101011 = 1 x 25 + 0 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20
= 32 + 0 + 8 + 0 + 2 + 1
= 43
When we add several binary numbers, we sometimes add the
carry 1 to a column that is not an adjacent column.
1110
1101
1100
1011
=
=
=
=
14
13
12
11
110010 = 50
We
We
We
We
added
added
added
added
column
column
column
column
0’s
1’s
2’s
3’s
carry
carry
carry
carry
to
to
to
to
column
column
column
column
1
2
4
5
Unless we add together many, many base 10 numbers we only
add carrys to adjacent columns.
Subtracting Binary Numbers
100 = 4
- 11 = 3
1 = 1
1100 = 12
-0111 = 7
? = ?
We borrow “1” from column 3 and add it to column 2
01100 = 12
-0111 = 7
? = ?
Next we borrow “1” from column 2 (11) add it to column 1
010100 = 12
-0111 = 7
? = ?
Finally, we borrow “1” from column 1 (10) and add it to
column 0. We now do the subtraction
010110 = 12
-0111 = 7
0101 = 5
Last Subtraction Example
100000
-001010
?????0
Position 4 borrows the 1 at position 5. The 0 at position 4
becomes 10.
0100000
-0 01010
0 ????0
Position 3 borrows 1 from the 10 at position 4 giving us
0110000
-00 1010
01 ????0
Position 2 borrows 1 from the 10 at position 3 giving us
0111000
-001 010
01? ??0
Position 1 borrows 1 from the 10 at position 2 giving us
0111100
-0010 10
0101 ?0
We can now get an answer,
0111100
-0010 10
0101 10
Writing the original problem we have,
100000
-001010
010110
Fractions
Fractions as ratios as integers
1/7, 2/5, 3/4, etc.
Fractions as decimal fractions
1/7 = 0.142857...
2/5 = 0.4
3/4 = 0.75
Terminating Decimal Fractions
2/5 and 3/4 generate terminating decimal fractions
Repeating Decimal Fractions
1/7 generates a repeating, non-terminating decimal fraction
Repeating Decimal Fraction Criteria
A fraction – the ratio of 2 integers – generates a nonterminating, repeating decimal fraction if and only if at
least one of the denominator’s prime factors is a prime
number other than 2 or 5.
Terminating Decimal Fraction Criteria
A fraction – the ratio of 2 integers – generates a
terminating decimal fraction if and only if the
denominator’s prime factors are 2 and 5.
Ambiguous are
0.999... = 1
0.0999... = 0.1
0.00999... = 0.01
We will always write 1, 0.1 and 0.01 and not the repeating
fractions.
Write the following fractions as decimal fractions
3/7, 3/5, 2/11, 3/8
3/7
=
=
3/5 =
2/11 =
=
3/8 =
0.428571428571...
0.[428571]
0.6
0.1818...
0.[18]
0.375