Category 3 - Number Theory - Meet #3 - Study Guide by qjk18715

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									Category 3 - Number Theory - Meet #3 – Study Guide
Topics : Bases, scientific notation

Converting bases : Most of the world uses base ten most of the time. Questions in math class are in base ten
   even though we don't bother to point it out each problem. So let's first examine a base ten number that
   we are familiar with.

First the base is indicated by a subscript at the end of the numeral. When there is no subscript we assume the
    numeral is in base 10. Example 3456 = 345610.

But what is 345610 really? Using basic place value :
345610 = 3000 + 400 + 50 + 6
345610 = 3(1000) + 4(100) + 5(10) + 6(1)
345610 = 3(103) + 4(102) + 5(101) + 6(100) notice 3456 is written out will all powers of 10, hence BASE 10

If you work in any other base, you would follow the same idea as the last line above. For example :

23456 = 2(63) + 3(62) + 4(61) + 5(60) which in base ten =
23456 = 2(216) + 3(36) + 4(6) +5(1) = 423 +108 + 24 + 5 = 56010

Changing a number from some other base to base ten is fairy simple if you follow the example above. Note
   the units digit which is often called the "ones" digit is always the ones digit in any base. It is also the
   remainder when the numeral is divided by that base.
   345610÷10 has a remainder 6
   23456 ÷6 has a remainder of 5

To change a number from base ten into another base is slightly more tricky and their are several methods that
   work. Say you want to change 345610 into base 4.
You can start by finding the largest power of 4 less than 3456. Powers of 4 are 1, 4, 16, 64, 256, 1024, 4096.
   So the largest is 1024 = 45. The largest multiple of 1024 less than 3456 is 1024x3 = 3072. 3 will be the
   first digit in our new numeral.
Now we look at 3456 - 3072 = 384. The largest power of 4 less than 384 is 256 and the largest multiple of
   256 less than 384 is 256 x 1, so 1 is our next digit.
Now we look at 384 – 256 = 128. The largest power of 4 less than(or equal to) 128 is 64. The largest
   multiple of 64 is 64 x 2 =128, so 2 is our next digit.
Now we look at 128 – 128 = 0. Since we have nothing left, the rest of the digits must be zero. But how
   many digits are left? Well we didn't use these powers of 4 yet :1, 4, or 16 so there will be three zeros.

So 345610 = 3120004

The easier way to do this shows less understanding, but is much more efficient. We will divide 3456 by 4
   repeatedly only keeping track of the remainders.
 864r 0      216r 0      54r 0     13r 2     3r1       0r 3
4 3456 ⇒ 4 864 ⇒ 4 216 ⇒ 4 54 ⇒ 4 13 ⇒ 4 3                  Now look at the remainders and use them as

   your digits starting with the right most digit and work to the left. 3120004
Category 3 - Number Theory -– Study Guide continued
Once you can convert back and forth you can add, subtract, multiply, and divide in other bases by changing
   to base 10 and then back again. But if you can think in other bases you can do the operations without
   changing bases. Note that the largest digit in base 10 is 9, in base 6 the largest digit is 5, always one less
   than the base. That makes sense because if you were working in base 4, then the third place value would
   be 42 = 16. If you put a 4 in that place, it would be equal to 4(42) = 4(16) = 64 which is actually the next
   place value over and therefore you would "carry" like we do normally in base 10.

If you do not know the base but you know the value of the number in another base you can write an equation
    to solve for the base. For example, if 53x = 38, you can write the equation 5(x) + 3 = 38 and then solve to
    find x = 7! If you had 324x you could write the expression 3x2 + 2x + 4.

Last note, if you are working in a base greater than 10 you need to start using letters as digits. In base 16, the
   digits are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F where F=15. So AF in base 16 = 10(16) +15 = 175

Scientific Notation
Scientific notation is a way to express really large or really small numbers without using so many digits.
    Instead you multiply by powers of 10 to make the number as big or small as you want.

Any number in scientific notation will be in the form N x 10p where 1 ≤ N < 10 in other words N has one
   digit before the decimal.

so 7,200,000 could be written as 7.2 x 106 since if we multiplied 7.2 by 10 six times you'd get 7,200,000(of
    that the decimal needs to be moved 6 places to the right)

also .0000341 could be written as 3.41 x 10-5 since 10-5 =.00001 and 3.41x.00001 = .0000341(or that the
    decimal needs to be moved five places to the left)

If you multiply two numbers in scientific notation you can work with the number of power parts separately
    first and then worry about adjusting to make sure the number is still in scientific notation.

(4.5x106)(6.2x107) = (4.5x6.2)(106x107) = 27.9 x 1013 but that is not in scientific notation anymore since 27.9
    is greater than 10. So we change 27.9 into 2.79(making is smaller by a factor of 10) and change 1013 to
    1014(making it bigger by a factor of 10). So (4.5x106)(6.2x107) = 2.79 x 1014

                          4.5 × 10 8 4.5 10 8
To divide work similarly           3
                                     =    × 3 = .625 × 10 5 BUT .625 is less than 1, so this is not in
                          7.2 × 10     7.2 10
   scientific notation anymore. We must change .625 into 6.25(make it bigger by a factor of 10), so we
                                                                              4.5 × 10 8
   must change 105 to 104 (making it smaller by a factor of 10). Giving us :           3
                                                                                         = 6.25 × 10 4
                                                                              7.2 × 10

You may need to practice you simplifying of exponents before work on the harder problems.

Especially note that a negative exponent means to put the flip that power to the other side of the fraction!

								
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