03FebComputing Logic
Covers exponets
3^4 = 333*3
Talks about any number to the 0 power = 1
A.e. 3^0 =1
A.e 257^0=1
B^n * B^m =B(n+m)
(3^2)^3= 3^6
(2^2)(3^2) =(23)^2
(x^3)(y^3)=(xy)^3
A fraction or quotient (a/b) raised to the power n is the same as the power applied to both numerator and denominator individually
(a/b)^n = (a^n)/(b^n)
Zero power proof:
Show that b^- = 1, for all b<>0
b^1/b^1 = 1 (any number divided by itself is 1)
b^1/b^1 = b^(1-1) = b^0 (subtract exponents)
So, b^0 = 1
First power proof:
Basically any number multiplied 1 time is one.
Reciprocal of a number is 1 divided by that number
Reciprocal of b is 1/b
Exception is 0: division by 0 is undefined
A number multiplied by its reciprocal is 1
Multiplying a number by its reciprocal is the same as dividing by that number
The reciprocal of a fraction is the switch of places of the numerator and denominator.
Reciprocal of a number b is b^-1
Reciprocal of 1000 is 1/1000 or 1000^-1
Reciprocal of a number with a positive exponent is a result with a negative exponent
Reciprocal of a number with a negative exponent is a result with a positive exponent #
click to edit
Basically reciprocal is switching from positive to negative or vise versa
10^3*10^-2 = 10^3+-2 = 10^(3-2) = 10^1 = 10
Create reciprocals by flipping exponents
Reciprocal of 3^2 is 3^-2
Reciprocal of 5^-3 is 5^3
Keep the same value, but chang the sign of the exponent by double flipping
5^-2 is the same as 1/5^2
x^-5 is the same as 1/x^-5
(-5)^2 = (-5)*(-5) = 25
Roots and Fractional Exponents
Invers operation undoes pervious operation
Taking a root is the invers of raising a number to a power
Taking square root undoes squaringa number
Taking cube root undoes cubing a number
Fractional exponents represent roots
/2 can be written as 2^(1/2)
Problem solving with exponents
10^6 is 1 with 6 zeroes after it
10^-2 is 1 divided by 100 (1 with 2 zeroes after it
Product of
Decimal value between 1 and 10 (not 10)
A power of 10
The decimal value contains significant digits (click on subpoints to see due to italics)
2500 is 2.51000, or 2.5 10^3
17500 is 1.75 1000, or 1.75 10^4
0.000027 is 2.7 * 10^-5 (count decimal to first sd
-0.000027 is -2.7 * 10^-5
sd = significant digits
Significant digits
All numbers in the decimal part of scientific notation are significant
This allows you to express accuracy
2.0 * 10^3 implies 2000 is known to 2 sd
2.000 *10^3 implies 2000 is known to all 4 sd
2.5* 10^-3 implies 0.0025 is known to 2 sd
2.500*10^-3 implies 0.0025 is known to all 4 sd
Significant digits example was saying something is 20 dollars or 19 dollars and 93 cents.. you basically just say 20 dollars because the dollar amount over the cent amount is more significant
When mailing an envelope the 1 oz range is what is significant, not the .00000001 of an oz
Study Materials
Chapter Summary
Rules of exponents
Reciprocals
Roots
Review questions
Problem solutions to even-numbered exercises