Download 02/08

Why
537
5 x
5 x
5 x
5 x
5 x
6 x
603
we carry numbers
+ 66 =
100 + 3 x 10 + 7 x 1 + 6 x 10 + 6 x 1 =
100 + 9 x 10 + 13 x 1 =
100 + 9 x 10 + 1 x 10 + 3 x 1 =
100 + 10 x 10 + 3 x 1 =
100 + 1 x 100 + 3 x 1 =
100 + 3 x 1 =
negative numbers
we can either see subtraction or the addition of negative
numbers.
a negative number is –1 x a positive number
subtraction is addition of a negative number
-23 + (-15) – 12 =
(-1)23 + (-1)15 + (-1)12 =
(-1)(23 + 15 + 12)
23 + 15 + 12 =
2 x 10 + 3 x 1 + 1 x 10 + 5 x 1 + 1 x 10 + 2 x 1 =
4 x 10 + 10 x 1 =
4 x 10 + 1 x 10 =
5 x 10 =
50
-23 + (-15) – 12 = -50
2(-4 + 3 x 2) =
2(-4 + 6) =
2(2) =
4
-3 – 5 =
(-1)3 + (-1)5 =
(-1)(3 + 5) =
(-1)8 =
-8
(-1)3 – (-1)5 =
(-1)3 + (-1)(-1)5 =
(-1)3 + 5 =
-3 + 5 =
2
exponents
Why does an x am = a(m+n)?
a3 x a4 = a7?
a3 = a x a x a
a4 = a x a x a x a
a3 x a4 = a x a x a x a x a x a x a
a3 x a4 = a7
Why does (am)n = amn?
(a2)3 = a6?
(a2)3 = (a2) x (a2) x (a2)
= (a x a) x (a x a) x (a x a)
= a x a x a x a x a x a
= a6
negative exponents
By definition, 1/an = a-n
Why does am/an = a(m-n)?
case 1: m > n
a5/a2 = (a x a x a x a x a)/(a x a)
= (a/a) x (a/a) x a x a x a
= a x a x a
= a3
case 2: m = n
a3/a3 = (a x a x a) / (a x a x a)
= (a/a) x (a/a) x (a/a)
= 1 x 1 x 1
= 1
case 2’: zero exponent
Why does a0 = 1?
a5/a5 = (a x a x a x a x a)/(a x a x a x a x a)
a5/a5 = a5 x 1/a5
a5 x 1/a5 =
a5 x a-5 = a5-5
a5-5 = a0
Since a5/a5 = a0,
a0 = (a x a x a x a x a)/(a x a x a x a x a)
= (a/a) x (a/a) x (a/a) x (a/a) x (a/a)
= 1 x 1 x 1 x 1 x 1
= 1
case 3: m < n
a2/a5 = (a x a)/(a x a x a x a x a)
= (a/a) x (a/a) x 1/(a x a x a)
= 1 x 1 x 1/(a x a x a)
= 1/a3
= a-3
Scientific Notation
nn...n = n.nnnn x 10k
1776 = 177.6 x 101
= 17.76 x 102
= 1.776 x 103
= 0.1776 x 104
= 0.01776 x 105
0.01776 =
=
=
=
=
0.1776 x 10-1
1.776 x 10-2
17.76 x 10-3
177.6 x 10-4
1776 x 10-5
Error Analysis
Rules for Significant Digits
“The digits that define a numerical value are called
significant digits. For a number with no decimal part, the
significant digits can be counted starting at the left and
counting up to the last nonzero digit. That gives the
minimum number of significant digits.”
Why minimum?
Are the zeroes the result of accurate measurement or
estimation?
Absolute Error
“The absolute error is the (absolute value of the)
difference between a measurement and the exact value.”
Absolute Error = |measurement – exact value|
The absolute error is not as important as the relative
error.
Relative Error
“The relative error is the ratio that results when the
absolute error is divided by the exact measurement. We
usually express this as a percentage.”
Relative Error = Absolute Error/exact value
= |measurement – exact value| / exact value
Dimensional Analysis
10 miles / hour =
10 miles / hour x 5280 feet/mile x 1 hour/ 3600 second
= (10 x 5280/3600) feet/second
= 1 x 101 x 5.28 x 103 /(3.6 x 103) feet/second
= 5.28 x 104/(3.6 x 103) feet/second
= 5.28/3.6 x 104 x 10-3 feet/second
~ 1.47 x 101 feet/second
~ 14.7 feet/second
To convert from one set of units to another we have
multiply by 1 many times. Multiplying by 1 cannot change
the value of any expression. Where is the 1?
1
1
1
1
1
=
=
=
=
=
5280 feet/mile
3 feet / 1 yard
1 hour /3600 seconds
1 hour / 60 minutes
1 minute/ 60 seconds