Numbers By Katie Hutton We can use numbers and math to describe things all around us. Numbers are everywhere. We use them everyday in everything. Perfect numbers What is a number? Real and Imaginary numbers 1 2 3 4 5 6 7 8 9 0 Numbers have been used for thousands of years, and there are many different properties of numbers that have been discovered Today's topics are: real and imaginary numbers, complex numbers, rational and irrational numbers, prime numbers, perfect numbers, and amicable numbers Perfect numbers are numbers that the sum of all of it's divisors, excluding itself, equal itself. 6 is a perfect number because 6 is divisible by 1,2,3, and 6, and 1+2+3=6 This is extremely rare and was discovered as early as 100 AD The next perfect numbers are

28=1 + 2 + 4 + 7 + 14

496=1+2+4+8+16+31+62+124+248

8121=1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016+ 2032 + 4064

After that they go to numbers like 33,550,336 and larger The 20th perfect number has over 5 thousand digits in it An arithmetical value, expressed by a word, symbol, or figure representing a particular quantity. Official Definition: There are different kinds of numbers that can be categorized into different areas. Some numbers have fascinating rules, while others are just numbers. Some numbers don't even "technically exist." Positive and negative numbers Numbers can be split into 2 groups, negative and positive 0 is the turning point for positive and negative numbers. Numbers to the left of 0 are negative and numbers to the right are positive. All of the numbers we have are made up of the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 Real numbers include all kinds of numbers Imaginary numbers are numbers that don't exist. There are real and imaginary numbers. Real numbers are numbers that we can see and understand, while imaginary numbers are numbers that we know must be there, but we don't know what they are Real Imaginary Almost every number, whether it be rational or irrational, a fraction, or a whole number, is a real number. We can see and easily identify real numbers Rational numbers are: Irrational numbers Irrational numbers are numbers that in decimal form do not stop or repeat in a pattern They go on forever with no noticeable pattern Some examples of irrational numbers are pi,

the square root of 2, and many others. They also can't be expressed as a ratio. 3.14159265358979323846264338327950288419716939937... 1.4142135623730950488016887242096980785696718753769480731766797379907324784621... Numbers that can be expressed by a ratio, include whole numbers, integers, and natural numbers as a decimal will repeat or end. Numbers like 4.67, 9, 3/4, or 3.33repeating are rational numbers Integers Integers are whole numbers They are not fractions or decimals and can be positive or negative 1, 900,

-123456789= :) 1/2, 5.564,

-12.3456789= :( They are represented by the letter i Instances where imaginary numbers are used, are when squaring negative numbers i x i=-1 If i squared equals -1 then you can do a lot more equations Ex. Now you know what the square root of -9 is Complex numbers Complex numbers have 2 parts, a "real" part and an "imaginary" part The standard formula is a+bi Complex numbers are binomials and are added and subtracted in similar ways. Binomials: an algebraic expression of the sum or the difference of two terms. Simplify (2 + 3i) + (1 – 6i).

(2 + 3i) + (1 – 6i) =

(2 + 1) + (3i – 6i) =

3 + (–3i) =

3 – 3i An example of a problem with complex numbers PRIME Numbers Whole Numbers wait for it..... the numbers 0,1,2,3,4,5 and so on. These Include 0. Natural Numbers sometimes known as counting numbers the numbers 1,2,3,4,5....... NO Z E R O Basically whole numbers without the 0 BYE ZERO :( Numbers that have no positive divisor other than 1 and itself. 3 is a prime number because it can only be divided by 1 and itself. 6 is not a prime number because it can be divided by 1, 2, 3, and itself Numbers that aren't prime are composite numbers, like 6. Amicable numbers Known in pairs Amicable means having a spirit of friendliness, so the numbers are "friendly" to each other. The first pair of amicable numbers are 220 and 284 all the natural divisors of 220, excluding 220 itself, add up to 284. 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 All the natural divisors of 284, excluding 284, add up to 220 1 + 2 + 4 +71 + 142 = 220 The pair was originally found by Pythagoras Amicable numbers= the divisors of one number excluding itself add up to the other number, and it works the other way around. Now for some problems..... Simplify. 1. -225 2. (15+5i)-(2i+13) copy these on a piece of paper. 3. (25+8i)+(5-10i) to multiply complex numbers you use foil First

Outer

Inner

Last (a+bi)(c+di) = ac + adi + bci + bdi2 4. (9+5i)(2+3i) Answer...

1. 15i 2. (15+5i)-(2i+13)=

(15+13)-(2i+5i)=

28-7i 3.

(25x8i)+(5-10i)=

(-10x8i)+(5+25)=

-80i+30 4.

(9+5i)(2+3i)=

9x2+9x3i+5ix2+5ix3i=

18+27i+10i+15i2=

18+27i+10i-15=

3+37i

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