• Like
Binomial Expansion Reflection
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

Binomial Expansion Reflection

  • 1,003 views
Published

 

  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Be the first to comment
    Be the first to like this
No Downloads

Views

Total Views
1,003
On SlideShare
0
From Embeds
0
Number of Embeds
1

Actions

Shares
Downloads
12
Comments
0
Likes
0

Embeds 0

No embeds

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
    No notes for slide

Transcript

  • 1. Binomial Expansion Reflection
    Hossam Khattab, Grade 8B
    Qatar Academy
    November 3rd, 2010
  • 2. Background Information
    Our guiding or main questions was “is there an easy way to do 0.992?”
    We discovered we could do this through binomial expansion
    0.992= (1-0.01) (1-0.01)
    = 12-2x1-0.01+(-0.01) (-0.01)
    = 0.9801
    This method is much quicker, and less hassled than using long multiplication
  • 3. General Rules
    For the square of the sum of two number we developed:
    (a+b)2 = a2 + 2ab + b2
    -This means the first term squared, plus the product of the first and second term times two, plus the last term squared
    For the square of the difference of two numbers we developed:
    (a-b)2 = a2 - 2ab + b2
    -This is slightly different, where the first term is still squared, however the product of the first and second term squared is then subtracted from the first term squared, then the last term squared is added
  • 4. Advantages as Opposed to Traditional Multiplication
    If you were an engineer 100 years ago, explain how our method may have been useful rather than just long multiplication?
    An engineer 100 years ago could rely on this method as a shortcut. Since an engineer 100 years ago would not have had a calculator they would be using pencil and paper.
    Shortcuts like this would therefore be important, as they increase efficiency and time productivity
    This is especially important to someone like an engineer who is bound by a time schedule or a limit
  • 5. To an Engineer….
    For someone like an engineer, this method is:
    Useful, because it is very straightforward, and has many applications
    Reliable, because it has little steps, with less room for error. For an engineer reliability is important
    Because it has a lower chance of error, it is accurate, and in engineering like building houses, it needs to be, because people are depending on this
  • 6. Long Multiplication vs. Binomial Expansion (992)
    Binomial Expansion
    992= (100-1)2
    = 10000 – 2x100x1 + 1
    = 9801

    Long Multiplication
    992=
    899
    x99
    891
    +8910
    ✗ 9801
  • 7. Explanation
    This method lets us write the number as either the sum or difference of two numbers
    The method allows you to multiply numbers you are comfortable with, in this case, 100 and 1.
    Alternative to long multiplication, because the method is different, so it is shorter.
    This is because writing the number as a sum or difference you end up with the same result, but during the process the multiplication is different
  • 8. Further Explained…
    In the case shown previously, binomial expansion is shorter and easier, because the number is very close to a hundred
    The method is useful because is takes less time, and is easier to do therefore less prone to error
    If you are not using the algebraic method, you are still able to use this method, by following the distributive law
  • 9. However…
    In some situations, our method becomes cumbersome and messy such as:
    When the numbers have many decimal places. This added to the whole numbers already to the left can get confusing
    At the end, it is harder to know where to place the decimal point
    Also, there are decimals which may have zeros, which are normally easily to multiply, but will later on give you more complicated decimals to add and multiply
  • 10. Examples
    96.022= (90+6+0.02)2
    = 902+90x6+90x0.02+6x90+ 62+6x0.02+0.02x90+0.02 x6+0.022
    = 8100+540+1.8+540+36 +0.12+1.8+0.12+0.04
    = 9219.8404
    96.02
    x96.02
    19204
    5461200
    86418000
    9219.8404
  • 11. Further Examples
    83.42= (80+3+0.4)2
    = 802+80x3+80x0.4+3x80+32 +3x0.4+0.4x80+0.4x3+ 0.42
    = 6400+240+32+240+9+0.12+32+0.12+0.16
    = 6955.56
    83.4
    x83.4
    3336
    25020
    667200
    6955.56
  • 12. Other Situations
    Other situations where using long multiplication is probably a better option than binomial expansion
    Number to powers greater than two, i.e. cubed, to the power four, etc.
    When numbers must be broken up into three ore more parts, such as three digit numbers
    When multiplying three or more two-digit numbers
    Numbers that have 3 or more digits
  • 13. Limitations Explained
    When numbers start getting into 4, or even just 3 digits, this method becomes hard, and defeats the purpose of the mental math, because it will involve complex additions, and multiplications, and you will probably end up using long multiplication to calculate within the original calculation
    When the number has to be split into more parts, the algebraic rule cannot work. Therefore you must use regular multiplication and expansion. You end up in turn, having to multiply every number by every number. This increases greatly every time you add a single digit to either of the two numbers involved.
  • 14. Examples
    (23)(21)(15)= (20+3)(20+1)(10+5)
    = 202+202+20x10+20x1+20x5+3x20+3x1+3x10+3x5
    = 400+400+200+20+100+60+3+30+15
    =7245
    23
    x21
    23
    460
    483
    x15
    2415
    4830
    7245
  • 15. Further Examples Pt. 2
    523= (50+2) (50+2) (50+2)
    = 503+50x2x2+2x50+2x50+2x2x2
    = 125000+200+100+100+8
    = 140608
    52
    x52
    104
    2600
    2704
    x52
    5408
    135200
    140608
  • 16. Pascal’s
    T
    R
    I
    A
    N
    G
    L
    E
    • Is formed by adding adjacent numbers and writing the answer in the line under in a brick-fashion
    • 17. First diagonal is formed from ones. Second is formed from counting numbers and the third is formed from triangular number. The white area is tetrahedral numbers
    • 18. Horizontal Sums are equal to two to the power of the row. This is a very useful as an alternate method to finding large powers, instead of both long multiplication and binomial expansion
  • Conclusion
    Binomial Expansion is very useful in general for multiplying 2 or 3 digit numbers, and squaring them, which would usually be difficult or would require long multiplication
    It is a shortcut method, to reduce working for products where long multiplication would otherwise be necessary, however it cannot completely replace long multiplication, simply because it starts to get confusing with many decimals, large numbers, and larger powers.