Mathematics - Review of Algebra Basic Concepts Flashcards by Guy Ben Arie

Q

Defining types of numbers: Natural numbers (N)

A

Numbers used for counting and ordering (6 coins, 3rd largest organ)

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:

Whole numbers

A

Natural numbers including zero (0,1,2,3..)

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:

Integers (Z)

A

Positive and negative counting numbers including zero (..-3,-2,-1,0,1,2..)

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:
Rational numbers(Q)

A

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)

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:
Real numbers ( R )

A

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.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:
Irrational numbers (I)

A

Real numbers that are not rational. (pi 3.1415.., √2)

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:

Imaginary numbers

A

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.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Defining types of numbers:
Complex numbers( C )

A

Includes real numbers, imaginary numbers, and sums and differences of real and imaginary numbers

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is a Prime Number?

A

A number >1 that is only divisible by itself or 1 to give a whole number. (3,5,7..)

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is an Exponent?

A

Any integer (n) that is representing the times of multiplication of a number by itself.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is a Power of ten?

A

Any integer power (n) of the number 10. For Example: 10^3 = 1000.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Where are Power of tens useful?

A

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 .

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What are the mathematical manipulations possible for Exponents? (Exponential Identities with Tens)
**3 Examples

A

Summations: (10^8) * (10^-2) = 10^6
Subtractions: (10^5) / (10^-5) = 10^10
Multiplications: (10^3)^3 = 10^9

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Order of operations - Complete the Rules:

Parenthesis ( ()/{}/[] ) comes _____

A

Parenthesis ( ()/{}/[] ) comes FIRST.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Order of operations - Complete the Rules:

Exponents comes ______

A

Exponents comes SECOND.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Order of operations - Complete the Rules:

Multiplication and division are ______

A

Multiplication and division are THIRD.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Order of operations - Complete the Rules:

Addition and subtractions are done ____

A

Addition and subtractions are done LAST (fourth).

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Order of operations - Complete the Rules:

Addition and subtractions are done last unless they are ____ __________ then they have higher priority

A

Addition and subtractions are done last unless they are INSIDE PARENTHESIS then they have higher priority

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

Order of operations - What are they ?

Now all of them together

A

1) Parenthesis come first
2) Exponents come seconds
3) Multiplication and division are third
4) Last are Addition and subtractions

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is Logarithm?

A

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 .

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is a Common Logarithm?

A

Common logarithm is the inverse of the ten-based exponentiation (powers of ten). The symbol of common logarithm can be „log10”, „ lg” or „log”.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is the inverse of the logarithmic calculation [lg0.01 = -2 ] in the domain of exponential calculations?
(This is just to demonstrate the logic)

A

[10^-2 = 0.01] is the inverse of [lg0.01 = -2 ].

This is because a Logarithm is the inverse mathematical operation of exponentiation.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is the inverse of the exponential calculation [10^0 = 1] in the domain of logarithmic calculations?
(This is just to demonstrate the logic)

A

[lg1=0] is the inverse of [10^0 = 1].

This is because a Logarithm is the inverse mathematical operation of exponentiation.

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

Q

What is a Natural logarithm?

A

Similar to common logarithm, but the base number is e (Euler’s number, e = 2.71828…) and its symbol is „ln” (logarithmus naturalis).

How well did you know this?

1

Not at all

2

3

4

5

Perfectly

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) = +/- √(2*c/a) (U may be better off writing the calculation down)

X(1,2) = +/- √(2*c/a) , 2t^2 +0t - 180 = 0 X(1,2) = +/- √(2*c/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(2π*f*t) 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(2π*f*t) y=sin(2π) y=sin(360) = 0 y=0

Geometric Shapes - Circle How can we calculate the Circumference(C)? (r = radius)

C=2*π*r

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 =4*π*r^2

Geometric Shapes - Sphere How can we calculate the Volume (V)? (r = radius) **This one is extra knowledge

V=4/3*π*r^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