Download - Ch. 1 Math Review(1)
Page 1
Math You Need to Know
Chapter 1 Circuits
ENGR 1375
2 Types of Numbers
• Exact Numbers – Precise to the exact number of digits presented
• 12 eggs in a dozen, not 12.2
• Used in text in descriptions, diagrams, and examples
• Approximate Numbers – ANY measurement obtained in laboratory settings where
the exact value cannot be found.
• Analog scale, digital display with only 2 digits instead of 4 (or
more).
NOTE: When working with exact and approximate numbers, care must be
made to include the appropriate number of significant digits in the result –
sometimes results must be rounded off to accomplish this. Page 2
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Powers of 10
• Examples of Powers of 10
100 = 1 x 100 = 1
101 = 1x 101 = 10
104 = 1x 104 = 1000
10-4 = 1x 10-4 = 0.0001 =
10-1 = 1x 10-1 = 0.1 =
10-6 = 1x 10-6 = 0.000001 =
Recall that 10-n =
1
10n
1
106
1
104 1
10
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Multiplying by of Powers of 10 • Multiplication by powers of 10: add exponents
103 x 104 = 107 (3 + 4 = 7)
101 x 107 = 108
101/2 x 103/2 = 104/2 = 102
103 x 10-4= 10-1 102 x 10-3/2 = 101/2
• To multiply a decimal form by a factor of 10,
move the decimal point to the right by the factor
100.0 x 103 = 100000 (move decimal right 3 places)
.01 x 105 = 1000 (move decimal right 5 places)
34.56 x 104 = 345600 (move decimal right 4 places)
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Divide by of Powers of 10
• Divide by powers of 10: subtract exponents 104 ÷ 102 = 102 (4 minus 2 = 2)
105 ÷ 10-3 = 108 (5 minus -3 = 8)
105/3 ÷ 10-1/3 = 106/3 = 102
• To divide a decimal form by a factor of 10, move the decimal point to the left by the factor 100 ÷ 104 = 0.01 (move to the left 4 digits)
1000000 ÷ 105 = 10 (move to the left 5 digits)
23400 ÷ 103 = 23.4 (move to the left 3 digits)
.034 ÷ 102 = 0.0034 (move to the left 2 digits)
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Combine multiply and divide
• Add or subtract exponents as necessary
• Combined with numbers, calculate numbers and
exponents separately
103 x 10-4
10-2 x 105
103 x 105 x 102
102 x 105 x 101 = 10 (3+5+2) - (2+5+1) = 102
= 10 (3-4) - (-2+5) = 10(-1 -3) = 10-4
2.1x103 x 4x105 x 3.3x102
5 x102 x 1.2 x105 x 2.4x101 x 2.1 x 4 x 3.3
5 x 1.2 x 2.4 =
= 1.925 x 102
10 (3+5+2) - (2+5+1)
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Exponents taken to powers
• When exponents are taken to powers,
multiply exponent times power (103 )4 = 10(3x4) = 1012
(10-3 )2 = 10(-3x2) = 10-6
(101/2 )4 = 10(1/2x4) = 102
( )3 = = 10-9
103
1
109
1
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Scientific Notation
• To express any real number in scientific notation,
put it into the form m x 10n , where m (mantissa)
is 1 or greater and less that 10 (1 ≤ m < 10). The
power, n, can be any integer value.
• Examples of scientific and floating point notation
46 294 = 4.6294 x 104 - (you must get mantissa between 1 and 10)
Move decimal place left 4 places (divide by 104) and insert factor 104
More examples
100 000 = 1 x 105 Floating point notation: 1 x E5
0.002 35 = 2.35 x 10-3 Floating point notation: 2.35 x E-3
468. Joules = 4.68x 102 J Floating point notation: 4.68 x E2
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Engineering Notation • Similar to Scientific notation (m x 10n ), except n is a
factor of 3 (3, 6, 9, -3, -6, or 0) and m can be any
convenient number less than 1000 (m < 1000)
• Examples: 477 x 103 , 0.0255 x 10-3 , 25.5 x 10-6
• Note that the 2nd and 3rd are the same value
• Note that there are equivalent prefixes for engineering
notation- See Table 1.2 in book
– Examples: kilo : 103 , mega: 106 giga: 109
milli: 10-3 , micro (also µ) 10-6 , pico: 10-9
• ANYTIME you see a prefix, you can mathematically
replace it with its exponent equivalent! Examples:
– 5 milliAmp = 5 x 10-3 Amp, 4.1 MegaOhm = 4.1 x 106 Ohms
• Or conversely, 25.4 x 103 Volt = 25.4 kiloVolt
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Changing Powers
• Anytime you need to change the exponent to another
value, you must also change the matissa.
– The rule is that if you multiply the 10 exponent by a factor of 10,
then you must divide the mantissa by the same factor
– Ex: Change 0.0255 x 10-3 to new mantissa with a power of 10-6
divide 10-3 by 103 to get 10-6), then must multiply the mantissa by
103 (move the decimal right by 3 places).
0.0255 x 10-3 = 25.5 x 10-6
Move decimal to right 3
places (mult by 103)
Move decimal to left 3
places ( divide by 103)
Changing Powers Example
• Convert 1500 µs (10-6 s) to ks (103 s)
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1500 x 10-6 s = 0.0000015 x 103 s
Move decimal to right 9
places ( multiply by 109)
Move decimal to left 9
places ( divide by 109)