Working with Imaginary Numbers (i)

  Imaginary Numbers are simply real numbers multiplied by the imaginary unit i which is representative of  √−1. The reason we use i as the imaginary unit is that we simply cannot multiply any number by itself to get a negative square, neither rationals, irrationals, negatives, nor zero will work, so we imagine a unit, i, that equals the square root of a negative (i = √−1). Now when we square i, we can get a negative number that is a square (i² = -1).

  Now that we understand the basic concept of i,  we can move on to its uses and other interesting properties. i's main use is to simplify a negative root, by basically pulling out the negative. So, the square root of -4 would be 2i! Find the root as though it was not negative then multiply it by the unit i. So simple. The unit i can be multiplied by integers, fractions, decimals, radicals, and can even be negative. We mainly use it in the quadratic equation, because quite often, we take square roots of negatives, thus we get an imaginary root.

  Now, look at this property of i. You could say it cycles between 4 numbers when you multiply i by i or take the powers of i. Observe:

• i × i = -1
• -1 × i = -i
• -i × i = 1
• then 1 × i = i again

Look at it as powers:

• i¹ = √−1or i
• i² = -1 
• i³ = -√−1or -i
• i⁴ = 1
• i⁵ =  √−1or i again

-Zech

Forming Committees

On every UIL Mathematics test, there seems to be a problem about forming a committee of people.

To form a committee, you'll be dealing with a combination. A committee of a selected group of people is always the same, even if the people are chosen in a different order. Therefore, order does matter, so you'll use the 'Choose' function, or nCr in your calculator.

EX: You want to form a committee of 3 people from a pool of 6 people. How many different combinations can you form?

ANSWER: 6 nCr 3, or 6 'Choose' 3, = 20. Therefore, there are 20 possibilities for a group of 3 people.

-Marisa

(Have an amazing Spring Break!)

GCD & LCM

The Greatest Common Divisor of two numbers is the largest integer that can divide two numbers with a remainder of 0. This can be found by writing a list of all divisors for each number and identifying which integers are divisors for both numbers and simply finding the largest common one, the last divisor they will have in common. (This can also be applied to polynomials) For example:

  The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.
  The divisors of 45 are 1, 3, 5, 9, 15, and 45.
  So their common divisors are 1, 3, 5, and 15. Thus, the GCD of 45 and 60 is 15!
(The GCD of two primes will be 1! But the GCD of a prime and it's multiple is the prime number.)

The Least Common Multiple of two numbers is the smallest integer that is a multiple of each number. This can also be found by writing a list of several multiples for each number and identifying inegers which are multiples for both numbers and simply finding the smallest one, the first one they have in common.

  The multiples of 15 are 15, 30, 45, 60, 75, 90, 105, 120...
  The multiples of 20 are 20, 40, 60, 80, 100, 120...
  So their common multiples are 60, 120, ... We look at the first (lowest) common multiple which is 60.

Here are some GCD and LCM examples to figure out:
GCD
1. 7 and 21
2. 33 and 121
3. 47 and 97
4. 47 and 235
5. 49 and 56

LCM
6. 9 and 12
7. 4 and 62
8. 17 and 21
9. 6 and 75
10. 18 and 24

Answers:
(1) 7, (2) 11, (3) 1, (4) 47, (5) 7, (6) 36, (7) 124, (8) 357, (9) 150, (10) 72

-Zech

Equidigital, Extravagant, and Frugal Numbers

Equidigital, Extravagant, and Frugal Numbers are all related in that they deal with the number of digits in their prime factorization and the nummber of digits of the number itself.


Extravagant numbers are the most numerous, for there are infinitely many of them. We are starting with extravagant numbers because they may be the simplest to understand and they encompass the vast majority of real, natural numbers, plus understanding extravagant numbers will help you understand equidigitals and frugal numbers. Extravagant numbers are also called wasteful numbers because they are natural numbers that have fewer digits themselves than the number of digits in their prime factorization. Think of them as wasteful in that they may have many large factors, but they waste their larger composite factors because those must be broken down into prime factors. Most, but not all composite numbers will be extravagant numbers. All extravagant numbers are composite because they are factorable, so, there are no prime extravagant numbers. The first few extravagant numbers are 4, 6, 8, 9, and 12, but not 10! To find out if a composite, natural number is extravagant, first find its prime factors:


4 = 2 * 2 = 2^2
6 = 2 * 3
8 = 2 * 2 *2 = 2^3
9 = 3 * 3 = 3^2
12 = 4 * 3 or 6 * 2 = 2 * 2 * 3


In the case of 4, 6, 8, and 9, they each have one digit themselves and are composed of two or more prime factors, making them extravagant. See, 12 may have 4 and 6 as factors, but they are composite and must be broken down further into prime factors. The number 12's prime factorization is 2 * 2 * 3, which makes three digits, 12 only has two digits, so it is extravagant. The composite number 10 is not extravagant though. Read on to Equidigital and Frugal Numbers to find out more...


Equidigital numbers are the next most numerous numbers because this set of numbers encompasses all prime numbers and many of the left over composite numbers that arent wasteful. Equidigital numbers are also known as economical numbers because they do not have to break down large composite factors (ie all their factors are prime already) and they have the same (equal) number of digits as does the number of digits in  their prime factorization. The first few equidigital numbers are 1, 2, 3, 5, 7, and 10. Take a look at their prime factors:


1 = 1
2 = 2
3 = 3
5 = 5
7= 7
10 = 2 * 5

1 is unique for it is neither prime nor composite, but a 'unit'. 1's prime factorization by definition is 1. The numbers 2, 3, 5, and 7 are prime, so they have already reached prime factorization, each one digit like the numbers themselves. Ten however is two digits in length and can be factored into 2 and 5, which makes two digits, so 10 is equidigital too! Read onward to find out about Frugal Numbers!

Frugal numbers, as you could assume by now are the least numerous set of numbers because this set of numbers more or less picks up what is left over from the sets of extravagant and equidigital numbers. You could also assume by now that frugal numbers are also known as economical numbers because they are natural numbers that do not have to break down large composite factors (again, all their factors are prime already), just like equidigital numbers. However, unlike equidigitals, frugal numbers have more digits by themselves than the number of digits in their prime factorization. There are no single digit or double digit frugal numbers and only 8 triple digit frugal numbers. They are 125, 128, 243, 256, 343, 512, 625, and 729. After this, frugal numbers become a bit more numerous: 1024, 1029, 1215, 1250, 1280...

125 = 5 * 5 * 5 = 5^3
128 = 2 * 2 * 2 * 2 * 2 * 2 * 2 = 2^7

Examples/Review:
Identify whether the following number is Equidigital (economical), Extravagant (wasteful), or Frugal (also economical).
1. 64
2. 1024
3. 24
4. 42
5. 25
6. 125
7. 1215
8.111
9. 12
10. 10

Answers: 1.) Equidigital 2.) Frugal 3.) Extravagant 4.) Extravagant 5.) Equidigital 6.) Frugal 7.) Frugal 8.) Equidigital 9.) Extravagant 10.) Equidigital

-Zech

Directrix

To find the Directrix of an equation, follow the following steps:
1. Write the equation of your parabola. For example, the equation might be (x- 12)^2 = 8 (y - 5)
2. Divide the coefficient in front of the parenthesis on the right side of the equation by 4. In this example, the result is 8/4 = 2
3. Subtract the number obtained from the coefficient in the parenthesis on the right side of the equation. In this example, 5 - 2 = 3
4. the number obtained is the Directrix!

-Josh

Roots

Example: If the roots of x^3 + bx^2 + cx + d = 0 are -1, 2, 5, then b + c + d equals:
One method could be to expand (x+1)(x-2)(x-5) and then add b,c and d but there is a simpler method of obtaining the answer. Let's start with b.

-b/a is the sum of the roots. since a is 1, -b= -1 + 2 + 5. b=-6

c/a is the result of adding the product of the roots taken two at a time. (-1)(2) + (-1)(5) + (2)(5) = c/a.
c =3

-d/a is the product of the roots. -d/a = (-1)(2)(5). d = 10

b + c + d = -6 + 3 + 10 = 7

-Kevin

Changing from Base 10 to Base 2

(This is Marisa!)
So I JUST discovered how to convert numbers from Base 10 to Base 2, which will occasionally show up on both the Mathematics and the Number Sense Tests! The easiest way to show you how to convert numbers is by solving a problem myself, step by step.


Let's say you would like to convert 46210 to ___2. First, find the largest multiple of 2 that can divide into 462. After that, begin your division!


1. 462/256 (Largest multiple of 2 that is less than 462) = 1, with a remainder, R, of 206. For each following step, you will take your remainder and divide it by the next smallest multiple of 2, until you reach zero.


2. 206/128 = 1R78.


3. 78/64 = 1R14.


4. 14 Cannot be divided by 32 or 16, but CAN be divided by 8. Therefore, remember the first two zeroes and divide by 8, continuing as normal. 14/8 = 1R6.


5. 6/4 = 1R2.


6. 2/2 = 1R0.


7. 0/1 = 0.


Once you've reached 0, go back and write down all of your answers, ignoring the remainders. Remember the two zeroes in step 4! You should end up with 111001110.


There you have it! To sum it up, 46210 is equal to 1110011102.