Mathematics - Review of Algebra Basic Concepts Flashcards
Defining types of numbers: Natural numbers (N)
Numbers used for counting and ordering (6 coins, 3rd largest organ)
Defining types of numbers:
Whole numbers
Natural numbers including zero (0,1,2,3..)
Defining types of numbers:
Integers (Z)
Positive and negative counting numbers including zero (..-3,-2,-1,0,1,2..)
Defining types of numbers: Rational numbers(Q)
Numbers that can be expressed as aratioof an integer to a non-zero integer. All integers are rational, but the converse is not true; there are rational numbers that are not integers.(3\2, 0.25)
Defining types of numbers: Real numbers ( R )
Numbers that can represent a distance along a line. They can be positive, negative, or zero. All rational numbers are real, but the converse is not true.
Defining types of numbers: Irrational numbers (I)
Real numbers that are not rational. (pi 3.1415.., √2)
Defining types of numbers:
Imaginary numbers
Numbers that equal the product of a real number and the square root of −1. Allows for “new” Algebra with √-1 instead of x. The number 0 is both real and imaginary.
Defining types of numbers: Complex numbers( C )
Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers
What is a Prime Number?
A number >1 that is only divisible by itself or 1 to give a whole number. (3,5,7..)
What is an Exponent?
Any integer (n) that is representing the times of multiplication of a number by itself.
What is a Power of ten?
Any integer power (n) of the number 10. For Example: 10^3 = 1000.
Where are Power of tens useful?
Power of tens are used when there are measurements of Biophysical quantities in very High/Low numbers. For Example the size of a Mitochondria could be 1.32 x 10^-6 .
What are the mathematical manipulations possible for Exponents? (Exponential Identities with Tens)
**3 Examples
Summations: (10^8) * (10^-2) = 10^6
Subtractions: (10^5) / (10^-5) = 10^10
Multiplications: (10^3)^3 = 10^9
Order of operations - Complete the Rules:
Parenthesis ( ()/{}/[] ) comes _____
Parenthesis ( ()/{}/[] ) comes FIRST.
Order of operations - Complete the Rules:
Exponents comes ______
Exponents comes SECOND.
Order of operations - Complete the Rules:
Multiplication and division are ______
Multiplication and division are THIRD.
Order of operations - Complete the Rules:
Addition and subtractions are done ____
Addition and subtractions are done LAST (fourth).
Order of operations - Complete the Rules:
Addition and subtractions are done last unless they are ____ __________ then they have higher priority
Addition and subtractions are done last unless they are INSIDE PARENTHESIS then they have higher priority
Order of operations - What are they ?
Now all of them together
1) Parenthesis come first
2) Exponents come seconds
3) Multiplication and division are third
4) Last are Addition and subtractions
What is Logarithm?
Logarithm is the inverse mathematical operation of exponentiation. It’s a mathematical question asking what is the power needed to get this number from the base I have. As in 10^? = 100, then ?=2 .
What is a Common Logarithm?
Common logarithm is the inverse of the ten-based exponentiation (powers of ten). The symbol of common logarithm can be „log10”, „ lg” or „log”.
What is the inverse of the logarithmic calculation [lg0.01 = -2 ] in the domain of exponential calculations?
(This is just to demonstrate the logic)
[10^-2 = 0.01] is the inverse of [lg0.01 = -2 ].
This is because a Logarithm is the inverse mathematical operation of exponentiation.
What is the inverse of the exponential calculation [10^0 = 1] in the domain of logarithmic calculations?
(This is just to demonstrate the logic)
[lg1=0] is the inverse of [10^0 = 1].
This is because a Logarithm is the inverse mathematical operation of exponentiation.
What is a Natural logarithm?
Similar to common logarithm, but the base number is e (Euler’s number, e = 2.71828…) and its symbol is „ln” (logarithmus naturalis).
ln(e) =
1
Since e^1 = e
What is the inverse of the logarithmic calculation [ln(a) = x] in the domain of exponential calculations?
(This is just to demonstrate the logic)
[e^x = a] is the inverse of [ln(a) = x].
This is because a Logarithm is the inverse mathematical operation of exponentiation.
ln (e^x ) =
x
Since e^x will give you e^x / Algebraically: ln (e^x)= x*ln(e) = x
What are the mathematical manipulations possible for Logarithmic identities ? (by the example of common logarithm) **Give 3 Examples
Summation: lg(25)+lg(4) = lg(4 * 25)= lg(100) = 2
Subtraction: lg(20)-lg(200 )= lg(20 / 200)= lg(0.1) = -1
Multiplication by num: 2 * lg(10) = lg(10^2) = lg(100) = 2
Why are equations crucial for Biophysics studies?
The laws of physics give the correlation between different quantities. The physical law can be treated as a mathematical equation, from which the unknown quantity (usual symbol: x) can be found.
What kind of equation is 4x+1=33?
What is the solution?
Linear equations with one variable(x).
4x+1=33 → 4x= 32 → x=32\4 → x=8
Linear equations with 1 unknown - In Biophysics
A race car moved a distance of X = 248 m with constant acceleration during the time interval of t = 5.78. Calculate the acceleration (a) of the race car. The physical law describing the distance-time correlation in case of constant acceleration is: X= (1/2)a(t^2)
(U may be better off writing the calculation down)
X= (1/2)a(t^2) → 2X= a*(t^2) → 2X/(t^2) = a
a = 248*2 / (5.78)^2 = 496 / 33.408 = 14.846.
The car has an acceleration of 14.846 m/(s^2)
**The Physics part of this will be fully explained later in the course
Quadratic equation with one variable(x)
These equations have 3 integers/parameters(a,b,c) that are built in general form of: ax^2 + bx + c = 0.
For [5x^2 − 26x = 24] Identify the a, b and c parameters of this equation:
a is 5
b is -26
c is -24
What kind of equation is 5x^2 − 26x = 24?
What is the formula helpful in the calculation of these kind of equation?
Quadratic equation with one variable(x)
The Quadratic formula:
X(1,2) = {-b +/- √(b^2 - 4ac)}/2a
What kind of equation is 5x^2 − 26x = 24?
Solve this equation with the Quadratic formula!
(U may be better off writing the calculation down)
Quadratic equation with one variable(x)
X(1,2) = {-b +/- √(b^2 - 4ac)}/2a
X(1,2) = {26 +/- √(676 + 480)}/10
X(1,2) = {26 +/- 34}/10
X1= (-8)/10 = -0.8 X2=60/10 = 6
A physical example: A race car moves with an acceleration of a = 4 m/s2. Calculate the amount of time needed the run a distance of x = 180 m. In this case, t is the unknown in the physical law:
X= (1/2)a(t^2).
**The rearrangement of this equation to the Quadratic forms yields: 2t^2 +0t - 180 = 0 and because b is 0, we can used the Simplified Quadratic formula:
X(1,2) = +/- √(2c/a)
(U may be better off writing the calculation down)
X(1,2) = +/- √(2c/a) , 2t^2 +0t - 180 = 0
X(1,2) = +/- √(2c/a)
t(1,2) =+/-√(360/4)
t(1,2) = +/- 9.486
Since time is only a positive forward going variable
t = + 9.486
**The Physics part of this will be fully explained later in the course
Two linear equations with two unknown variables (x and y )
What are the 3 possibilities for solving them?
Graphing (Will be shown in Class)
Substituting
Summation/Subtraction of Equations
In a Two linear equations with two unknown variables (x and y ) the only possible solution for x and y can be a solution that -
that satisfies BOTH equations!
*meaning x and y VALUES FOUND will fit correctly in each of the equations.
Solve these linear equations with Substituting:
Y=2X
X+Y=6
Y=2X, X+Y=6 X+2X=6 3X=6 X=2, Y=2*2=4 (2,4) satisfies BOTH equations!
Solve these linear equations with Summation/Subtraction:
X+Y=5
X-Y=-3
X+Y=5, X-Y=-3 X+Y+X-Y=5-3 2X=2 X=1, 1+Y=5 → Y=4 (1,4) satisfies BOTH equations!
What are the possible Trigonometric functions?
Sinus, Cosine, Tangent, Cotangent
What is a Sinus Function?
Basic Definition
It is a function that is applicable for every Right triangle (with 90 degrees), For a given angle (0→90) it will return the number that represents the ratio between the leg INFRONT of the angle and the hypotenuse.
What is a Cosine Function?
Basic Definition
It is a function that is applicable for every Right triangle (with 90 degrees), For a given angle (0→90) it will return the number that represents the ratio between the leg NEXT TO the angle and the hypotenuse.
What is a Tangent Function?
Basic Definition
It is a function that is applicable for every Right triangle (with 90 degrees), For a given angle (0→90) it will return the number that represents the ratio between the leg INFRONT of the angle and the leg NEXT TO the angle.
What is a Cotangent Function?
Basic Definition
It is a function that is applicable for every Right triangle (with 90 degrees), For a given angle (0→90) it will return the number that represents the ratio between the leg NEXT TO the angle and the leg INFRONT of the angle.
Trigonometric functions:
In a Right triangle, the Hypotenuse value is X, the given angle is Y degrees and the leg INFRONT of the angle is Z. How would you write the Trigonometric Function representing the relationship here? Write the equation!
Sin (Y) = Z/X
Trigonometric functions:
In a Right triangle, the Hypotenuse value is Z, the given angle is X degrees and the leg NEXT TO the angle is A. How would you write the Trigonometric Function representing the relationship here? Write the equation!
Cos (X) = A/Z
Trigonometric functions:
In a Right triangle, the Hypotenuse value is A, the given angle is B degrees and the leg NEXT TO the angle is C. How would you write the Trigonometric Function representing the relationship here? Write the equation!
Cos (B) = C/A
Trigonometric functions:
In a Right triangle, the Hypotenuse value is D, the given angle is E degrees and the leg INFRONT of the angle is F. How would you write the Trigonometric Function representing the relationship here? Write the equation!
Sin (E) = F/D
Trigonometric functions:
In a Right triangle, the the leg NEXT TO the angle is C, the given angle is Q degrees and the leg INFRONT of the angle is F. How would you write the Trigonometric Function representing the relationship here? Write the equation!
Tan (Q) = F/C
What is the value of x here:
[sin x = 1/2]
**Possible to calculate with the sinus-1 function with the calculator. You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 30 degrees
**This result is useful to memorize.
What is the value of x here:
[sin x = 1]
**Possible to calculate with the sinus-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 90 degrees
**This result is useful to memorize.
What is the value of x here:
[sin x = (√2)/2]
**Possible to calculate with the sinus-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 45 degrees
**This result is useful to memorize.
What is the value of x here:
[sin x = (√3)/2]
**Possible to calculate with the sinus-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 60 degrees
**This result is useful to memorize.
What is the value of x here:
[sin x = 0]
**Possible to calculate with the sinus-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 0 degrees
**This result is useful to memorize.
What is the value of x here:
[cos x = 0]
**Possible to calculate with the cos-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 90 degrees
**This result is useful to memorize.
What is the value of x here:
[cos x = 1/2]
**Possible to calculate with the cos-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 60 degrees
**This result is useful to memorize.
What is the value of x here:
[cos x = (√3)/2]
**Possible to calculate with the cos-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 30 degrees
**This result is useful to memorize.
What is the value of x here:
[cos x = (√2)/2]
**Possible to calculate with the cos-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 45 degrees
**This result is useful to memorize.
What is the value of x here:
[cos x = 1]
**Possible to calculate with the cos-1 function with the calculator.You can find the inverse sine function on the calculator by using the „INV + SIN” or „2ndF + SIN” button combinations.
x = 0 degrees
**This result is useful to memorize.
The calculator could be set (in Trigonometry) to Degrees or ______ . In this latter setting [sin x = 1/2] will be equal 0.524 ___
RADIANS
[sin x = 1/2] will be equal 0.524 rad
In Harmonic oscillation, the displacement of the body from equilibrium position is described with the ____ function.
In Harmonic oscillation, the displacement of the body from equilibrium position is described with the SINE function.
Biophysics Application of Trigonometry - Harmonic Oscillation: y= A sin(2πft)
Where A is the amplitude (maximum displacement) and f is the frequency (number of oscillation in a unit time).
For A=1 meter, f = 1Hz and t = 1 sec what will be the y (actual displacement) ?
(2π rad = 360 degrees)
y= A sin(2πft)
y=sin(2π)
y=sin(360) = 0
y=0
Geometric Shapes - Circle
How can we calculate the Circumference(C)?
(r = radius)
C=2πr
Geometric Shapes - Circle
How can we calculate the Surface area(A)?
(r = radius)
A = π*r^2
Geometric Shapes - Sphere
How can we calculate the Surface area(A)?
(r = radius)
A =4πr^2
Geometric Shapes - Sphere
How can we calculate the Volume (V)?
(r = radius)
**This one is extra knowledge
V=4/3πr^3
**This one is extra knowledge
The definition of Radian is α=i/r.
Where i is the length of a circular arc infront of the α angle, and r is the radius. if the i of a circle circumference is equal 2π (or 360 degrees), How much radians are there in a full circle (the irrational number)?
360 degrees = 2π rad = 6.28 rad
**π=3.14
If 360 degrees = 2π rad = 6.28 rad, how many degrees equal 1 radian?
1 rad = 360/2π = 57.3 degrees
If 360 degrees = 2π rad = 6.28 rad, how many radians equal 1 degree?
1 degree = 2π /360= 0.01745 rad
