Chapter 10 Physics & Math: Mathematics
hecto
h, 10²
kilo
k, 10³
logA A
1
1 electron volt to joule
1.602x10^-19 Joule
1 amu to kg
1.661x10^-27 kg
common logarithms
base-ten logs, log₁₀
centi
c, 10⁻²
division with exponents
in division, if both bases are the same, subtract the exponent of the denominator from the exponent of the numerator, e.g. X⁹/ X³= X⁶
√2
~1.4
cos 180
-1
log 1/A
-logA
logA 1
0
X⁻⁵
1/X⁵
1 cal to joule
4.184J
1lb to newton
4.45N
20²
400
1 mile to feet
5280 ft
giga
G, 10⁹
conversion between natural and common logs
log x= lnx/2.303
significant figures
provide an indication of our certainty of measurement when performing calculatinons. Determined by the precision of the instrument being used for measurement.
estimate value of 15.4 / 3.80
shift both numbers in the same direction, adjust the divisor (number you are dividing by) first. 16/4~ 4
(X/Y)⁴
X⁴/Y⁴
√3
~1.7
e
~2.718
X^ (2/3)
³√X²
sin 180
0
sin0
0
tan 180
0
mega
M, 10⁶
tetra
T, 10¹²
tan 90
undefined
tan0
0
X⁰
1
cos0
1
how to determine number of significant figures
1. count all numbers between first nonzero digit on left and last nonzero digit on right. Any digit in between is significant 2. any zeros to the left of the first nonzero digit are leading zeros, not significant 3. if there are zeros to the right of the last nonzero digit and there is a decimal point, they are significant 4. for measurements, last number is usually an approximation and is not significant
substitution for algebraic systems 5x-2y=11 3x+4y=17
1. solve for one variable, y=5x-11/2 2. insert expression into other equation, 3x+4(5x-11/2)=17 3. isolate the variable and solve resulting equation, x=3 4. solve for the other variable using that, y=2
algebraic systems equations by setting equations equal 5x-2y=11 3x+4y=17
1. solve for the same variable on both sides of the equation and then set the two equations equal to each other y=5x-11/2 and y=17-3x/4 2. set equations equal, isolate variable and solve for it, x=3 3. solve the other variable by plugging in, y=2
11²
121
12²
144
13²
169
14²
196
1 inch to cm
2.54 cm
15²
225
16²
256
17²
289
18²
324
1L to ounces
33.8 ounces
19²
361
log A^B
BlogA
determine cylinder volume with radius that is measured 7.45m and height 8.323m
V=A(base)* height. Radius has least number of significant digits, 2, so answer is 1.5*10^3
inverse relationships
an increase in one variable is associated with a proportional decrease in the other
speedometer registers at 35mph. What is the speed in meters per second?
convert distance measurements with dimensional analysis to get 56,327m/hr convert time measurements to get 15.6m/s
estimating logs
convert the log(number) to scientific notation. Log (nx10^m)=log(n)+log(10^m)= m+log(n). n is between 1-10, so its log will be a decimal between 0-10. The closer n is to 1, the closer log n will be to 0; the closer n is to 10, the closer logn will be to 1. Approximate that log (nx10^m) is m+0.n, e.g. log (9.2x10^8) is 8>0.92=8.92
deci
d, 10⁻¹
deka
da, 10¹
ejection fraction is proportional to left ventricular volume expelled with each contraction of heart. Patient has ejection fraction of .7, cardiac output is 5L/min, HR is 80bpm. What is volume of left ventricle in this person?
determine volume ejected per beat: 5L/min / 80bpm= .0625L/beat. Only 70% of volume is expelled per heart beat, so we can determine the volume expelled: .0625L/beat x .7/beat= .0625/.7= .0893L
solving square roots that are not perfect
divide a number into known squares in attempt to reduce it, e.g. √180= √4x √9x √5= 2x3x√5= 6√5. Then estimate by knowing that it is between 2 and 3 because those squares are 4 and 9. Estimate √5=2.2, then 6(2.2)=13.2
inverse trignometric fucntions
each trigonometric function also has an inverse function, e.g. sin⁻¹. Can use length measurements to find an angle, e.g. sin⁻¹(a/c)= angle
multiplication with exponents
exponents can be manipulated directly if the base number is the same. When multiplying two numbers with the same base, the exponents are added to determine the new number, e.g. X³+X⁶=X⁹
solving division with scientific notation
extend out to get a perfect square. Since radical is 1/2 exponent, multiply the exponent inside by 1/2, e.g. √(4.9x10^-7) = √49x10^-8=7x10^-4
(X³)⁴
for a number that is raised to an exponent and then raised again to another exponent, the two exponents are multiplied, X¹²
estimation in multiplication
for complex multiplication problems, e.g. (3.17x 10^4) x (4.53x10^5), round the decimal place back. If you round one answer up, round the other answer down to compensate. Estimation would be (3.2x10^4) x (4.5x10^5)
math with significant figures
for multiplication and division, maintain as many digits as possible, then round to have as many significant digits that is the same as the number with least amount of significant digits. For addition and subtraction, decimal points are maintained. At one end, leave as many numbers after decimal point as appear in the number with least numbers after decimal
direct relationships
increasing one variable proportionately increases the other; as one decreases, the other decreases by the same proprotion
log AxB
logA + logB
log A/B
logA - logB
natural logs
loge or ln
milli
m, 10⁻³
estimation in division
make proportional adjustments in the same direction, e.g. round both numbers up or both numbers down
scientific notation
method of writing numbers that takes advantage of a powers of ten. A number is written with a significand and an exponent
elimination to solve algebraic systems of equation 5x-2y=11 3x+4y=17
multiply or divide one (or both) of the equations to get the same coefficient in front of one of the variables in both equations. Then add or subtract equations as necessary to eliminate one of the variables. Solve for one variable, then use it to solve for the other variable
nano
n, 10⁻⁹
p
p can be shorthand for -log, e.g. pH= -log[H+]
pico
p, 10⁻¹²
significand
the number before the multiplication sign in scientific notation. Has an absolute value in the range [0,10). Cannot begin with 0 or have >1 digit before the decimal place
micro
u, 10⁻⁶
adding numbers with exponents
when adding or subtracting numbers with exponents, true value must be calculated before addition or subtraction can be performed, e.g. 3^2 + 3^2= 18, not 6^2