Math Notes

Open Top Box deconstruction notes

• Map of the Open Top Box with A = 1152
• The box end is:

• The box width = w - 2 (¼)

• V = w x h x d = (w -½ h) (¼ h) (¼ h)
• In order to fold into a box, w-½ must be positive w-½ h > 0 or w ½ h
• If A = 1152, the “w” w*h/w = 1152/w
• V = w x h x d =(w-½ h) (¼ h) (¼ h)
• V = (w-½ * 1152/w) (¼*1152/w) (¼*1152/w)
• = (w -576/w) (288/w) (288/w)

• We know that these sets of numbers exist, because we can ask questions that require them. If an operation never requires a new set of numbers, then we say the set is “closed”, or completely self-contained, under that operation

• “The set of natural numbers is closed with respect to addition.”
• The first blank is the name of a set of numbers. The second blank is the name of an operation.
• This means that if you pick any two natural numbers and add them, you will get a new natural number. “The set of natural numbers is NOT closed under subtraction.” Because 1 – 2 gives an answer that is NOT a natural number (-1 is not a natural number)

• The sets of natural and whole numbers are closed under addition and multiplication.
• The set of integers is closed under addition, subtraction, and multiplication
• The set of rational numbers is closed under addition, subtraction,multiplication*, and division*
• The set of real numbers is closed under addition, subtraction, multiplication*, and division*, but NOT exponents

• Given: y = ax2 + bx + c where a, b, and c are all real numbers
• Solve this equation: 0 = x2 + 1
• - 1 = x2
• X = +- square root -1
• X = ???

• For the set to be closed, the answer must be a number in the set
• So, what is that number?
• What is square root -1 ?
• Square root -1 = the thing that when you square it, you get -1
• There is NO such real number!
• So, we define square root -1 = i, or equivalently - 1 = i2
• This is the imaginary number
• If a number is some multiple of ݅, we call it an imaginary number:
• Each of these is an example of an imaginary number
• If a number is a combination of a real number and some multiple of ݅, we call it a complex number:  
• Each of these is an example of a complex number