Showing posts with label scientific notation. Show all posts
Showing posts with label scientific notation. Show all posts
Wednesday, June 17, 2015
Links to notes for 17 and 18 June 2015
Notes for percent and other scales based on powers of 10.
Practice for rates and scientific notation.
Links to all scientific notation posts.
Topics not covered in class that will be covered on Thursday and on homework 2.
Changing repeating decimals to fractions in lowest terms from 2012.
Changing repeating decimals to fractions in lowest terms from 2011. (Different practice problems.)
Thursday, June 21, 2012
The basics of scientific notation
Scientific notation can be used to represent any positive number. Usually, we only use it for really small numbers (close to zero) or really big numbers (more than a billion), but the method will work for anything.
Scientific notation is the form a x 10^b, where a is a number between 1 and 10 (1 is included but 10 isn't) and b is an integer, which means a whole number or the negative of a whole number.
The letter a is called the significand and b is the exponent or sometimes the order of magnitude.
Example with a "regular sized" number.
The number of feet in a mile is 5,280. I want to write it as a number between 1 and 10 times a power of 10. 5,280 is a four digit number, so the nearest power of 10 is 1,000 or 10^3. If I divide 5280/1000, I get 5.28. The mechanical way to do this is to move the decimal place on 5280. over three places to the left.
5280. =
528.0 x 10^1 =
52.80 x 10^2 =
5.280 x 10^3 or 5.28 x 10^3 (Your calculator will not write trailing zeros after a decimal.)
The last line is scientific notation because 1 < 5.28 < 10. 5280, 528 and 52.8 are too big to be significands.
One foot is 1/5280 of a mile. Type this into your calculator and you get 0.000189394..., a decimal which goes on beyond our calculators limits. Let's round this to 0.000189. This is called rounding to three significant digits, which means how many digits we have after we get past the leading zeros.
Because this is less than one, we are going to have to multiply by 10 raised to a negative exponent to get the scientific notation.
0.000189 =
0.00189 x 10^-1 =
0.0189 x 10^-2 =
0.189 x 10^-3 =
1.89 x 10^-4
We would say this number in scientific notation as "one point eight nine times ten to the negative fourth".
Your calculator only has ten places for digits, so a number like 25,000,000,000 (twenty five billion) is too big to be written in regular notation. If you type it in, the TI-30XIIs will display 2.5 x 10^10, or "two point five times ten to the tenth power".
Multiplying numbers in scientific notation
If I have two numbers in scientific notation, here is how I multiply them together.
1) Multiply the significands.
2) Add the exponents.
3) If the product from step 1 is more than 10, divide it by ten (move the decimal place one to the left) and add 1 to the exponent.
Example.
(6.1 x 10^7) x (4.3 x 10^9) This is 61,000,000 x 4,300,000,000.
1) 6.1 x 4.3 = 26.23
2) 7 + 9 = 16
3) This gives us 26.23 x 10^16. 26.23 is too big to be a significand, so we divide it by 10 and multiply 10^16x10, which gives us 10^17. The answer in scientific notation is 2.623 x 10^17.
Scientific notation is the form a x 10^b, where a is a number between 1 and 10 (1 is included but 10 isn't) and b is an integer, which means a whole number or the negative of a whole number.
The letter a is called the significand and b is the exponent or sometimes the order of magnitude.
Example with a "regular sized" number.
The number of feet in a mile is 5,280. I want to write it as a number between 1 and 10 times a power of 10. 5,280 is a four digit number, so the nearest power of 10 is 1,000 or 10^3. If I divide 5280/1000, I get 5.28. The mechanical way to do this is to move the decimal place on 5280. over three places to the left.
5280. =
528.0 x 10^1 =
52.80 x 10^2 =
5.280 x 10^3 or 5.28 x 10^3 (Your calculator will not write trailing zeros after a decimal.)
The last line is scientific notation because 1 < 5.28 < 10. 5280, 528 and 52.8 are too big to be significands.
One foot is 1/5280 of a mile. Type this into your calculator and you get 0.000189394..., a decimal which goes on beyond our calculators limits. Let's round this to 0.000189. This is called rounding to three significant digits, which means how many digits we have after we get past the leading zeros.
Because this is less than one, we are going to have to multiply by 10 raised to a negative exponent to get the scientific notation.
0.000189 =
0.00189 x 10^-1 =
0.0189 x 10^-2 =
0.189 x 10^-3 =
1.89 x 10^-4
We would say this number in scientific notation as "one point eight nine times ten to the negative fourth".
Your calculator only has ten places for digits, so a number like 25,000,000,000 (twenty five billion) is too big to be written in regular notation. If you type it in, the TI-30XIIs will display 2.5 x 10^10, or "two point five times ten to the tenth power".
Multiplying numbers in scientific notation
If I have two numbers in scientific notation, here is how I multiply them together.
1) Multiply the significands.
2) Add the exponents.
3) If the product from step 1 is more than 10, divide it by ten (move the decimal place one to the left) and add 1 to the exponent.
Example.
(6.1 x 10^7) x (4.3 x 10^9) This is 61,000,000 x 4,300,000,000.
1) 6.1 x 4.3 = 26.23
2) 7 + 9 = 16
3) This gives us 26.23 x 10^16. 26.23 is too big to be a significand, so we divide it by 10 and multiply 10^16x10, which gives us 10^17. The answer in scientific notation is 2.623 x 10^17.
Friday, June 24, 2011
Practice for underflow and overflow in scientific notation.
Here is a link to a previous post about scientific notation, underflow and overflow.
Consider 80^80. If you enter this into your calculator, you will likely get an overflow error because the answer is more than 10^100. Here is how we can get around this, by splitting the number into two parts that are less than 10^100 and multiplying them together.
80^40 = 1.329227996 x 10^76
If we multiply 80^40 by 80^40, we will get 80^80. We need to square 1.329227996 to get the new significand, and the new exponent will be 10^(76+76) = 10^152
1.32922² = 1.766825808..., but since we only squared the number with six significant digits, we can only trust the answer to five significant digits, so our best answer is 1.7668 x 10^152.
More practice.
a) 40^80
b)40^-80
Answers in the comments.
Friday, February 4, 2011
Underflow and Overflow on your calculator
Really big numbers and numbers really close to zero are written on your calculator in scientific notation, a x 10^b, where b is an integer and a, known as the significand, is a number more than or equal to 1 but less than 10, which is to say exactly one digit to the left of the decimal place. Some calculators will write this in the form a E b. For example, it's about 96,000,000 miles to the sun. A mile is 5,280 feet and a foot is 12 inches. This means the number of inches to the sun has more than ten digits, so the TI-83 will write the answer as 6.08256E12, which is their way of writing 6.08256 x 10^12, which we could say in words as about six trillion.
With the exception of the high end calculator the TI-89, most calculators have decided they won't display numbers where the exponent is greater than 99 or less than -99. Some numbers in probability are larger than 10^100 or smaller than 10^-100, so the need can arise to express these. We can do this by using our calculators to find two number that multiply to the number we want that the calculator can represent, then multiplying those numbers together by hand. It's not that difficult to multiply numbers in scientific notation. You just add the exponents together to get the new exponent and multiply the significands. Multiplying two number less than 10 can give you a product more than ten. If that happens, you divide the product by ten (move the decimal over to the left) and add 1 to the exponent.
Simple example: 2 x 10^90 x 7 x 10^80 would be 14 x 10^170, but this isn't scientific notation because 14 > 10. To change it to scientific, we change 14 to 1.4 x 10, which raises the exponent by 1, so in scientific notation the answer is 1.4 x 10^171.
Overflow example: 80! is more than 10^100, so we need to split it up into two factors.
Factor #1: 50! = 50 x 49 x 48...x 3 x 2 x 1 = 3.0414 x 10^64
Factor #2: 80 nPr 30 = 80 x 79 x 78... x 53 x 52 x 51 = 2.3532 x 10^54
Multiplying the powers of 10 together is just 10^(64+54) = 10^118.
3.0414 x 2.3532 = 7.15702..., so rounding to four places total, which is called four significant digits, the final answer would be 80! = 7.157 x 10^118.
Underflow example: 1/80! is less than 10^-100, so we need to split it up into two factors.
Factor #1: 1/50! = 1/(50 x 49 x 48...x 3 x 2 x 1) = 3.2879 x 10^-65
Factor #2: 1/(80 nPr 30) = 1/(80 x 79 x 78... x 53 x 52 x 51) = 4.2496 x 10^-55
Multiplying the powers of 10 together is just 10^(-66+-55) = 10^-120.
3.2879 x 4.2496 = 13.96338..., which is more than 10, so we change it to 1.396338 x 10^1, combine 10^1 with 10^-120 to get 10^-119, and the final answer is 1/80! = 1.396 x 10^-119.
Practice problems:
Overflow: 90!
Underflow: 1/90!
Answers in the comments.
With the exception of the high end calculator the TI-89, most calculators have decided they won't display numbers where the exponent is greater than 99 or less than -99. Some numbers in probability are larger than 10^100 or smaller than 10^-100, so the need can arise to express these. We can do this by using our calculators to find two number that multiply to the number we want that the calculator can represent, then multiplying those numbers together by hand. It's not that difficult to multiply numbers in scientific notation. You just add the exponents together to get the new exponent and multiply the significands. Multiplying two number less than 10 can give you a product more than ten. If that happens, you divide the product by ten (move the decimal over to the left) and add 1 to the exponent.
Simple example: 2 x 10^90 x 7 x 10^80 would be 14 x 10^170, but this isn't scientific notation because 14 > 10. To change it to scientific, we change 14 to 1.4 x 10, which raises the exponent by 1, so in scientific notation the answer is 1.4 x 10^171.
Overflow example: 80! is more than 10^100, so we need to split it up into two factors.
Factor #1: 50! = 50 x 49 x 48...x 3 x 2 x 1 = 3.0414 x 10^64
Factor #2: 80 nPr 30 = 80 x 79 x 78... x 53 x 52 x 51 = 2.3532 x 10^54
Multiplying the powers of 10 together is just 10^(64+54) = 10^118.
3.0414 x 2.3532 = 7.15702..., so rounding to four places total, which is called four significant digits, the final answer would be 80! = 7.157 x 10^118.
Underflow example: 1/80! is less than 10^-100, so we need to split it up into two factors.
Factor #1: 1/50! = 1/(50 x 49 x 48...x 3 x 2 x 1) = 3.2879 x 10^-65
Factor #2: 1/(80 nPr 30) = 1/(80 x 79 x 78... x 53 x 52 x 51) = 4.2496 x 10^-55
Multiplying the powers of 10 together is just 10^(-66+-55) = 10^-120.
3.2879 x 4.2496 = 13.96338..., which is more than 10, so we change it to 1.396338 x 10^1, combine 10^1 with 10^-120 to get 10^-119, and the final answer is 1/80! = 1.396 x 10^-119.
Practice problems:
Overflow: 90!
Underflow: 1/90!
Answers in the comments.
Wednesday, July 1, 2009
Practice problems for the test on 7/2
The Transformers sequel made $108,966,307 this weekend.
a) Round this number to one significant digit.
b) Round this number to two significant digits.
c) Round this number to three significant digits.
According to the calculator 50! rounds to 3.041 x 10^64 and 60! rounds to 8.321 x 10^81, both rounded to four significant digits. Find the following values.
d) 50! / 60!
e) 60! / 50!
f) 50! x 60!
(note: the calculator will tell you overflow error on problem f because it doesn't like numbers bigger than 10^100 power. You will have to do this one by hand.)
Answers in the comments.
a) Round this number to one significant digit.
b) Round this number to two significant digits.
c) Round this number to three significant digits.
According to the calculator 50! rounds to 3.041 x 10^64 and 60! rounds to 8.321 x 10^81, both rounded to four significant digits. Find the following values.
d) 50! / 60!
e) 60! / 50!
f) 50! x 60!
(note: the calculator will tell you overflow error on problem f because it doesn't like numbers bigger than 10^100 power. You will have to do this one by hand.)
Answers in the comments.
Tuesday, June 30, 2009
Preview for 6/30
Today, the topics are going to be the metric system and the English measurement system and how to switch back and forth, and how to represent numbers using scientific notation, which is useful for really big numbers and numbers really close to zero. Scientific notation is used by calculators when there are too many digits to represent a number the regular way.
If we have time, the next topic will be logarithms.
There will be a quiz at the end of class. The topics include greatest common divisor (g.c.d.), least common multiple (l.c.m), representing numbers in decimal, hexadecimal and binary, and a basic understanding of the root words and prefixes of the metric system.
If we have time, the next topic will be logarithms.
There will be a quiz at the end of class. The topics include greatest common divisor (g.c.d.), least common multiple (l.c.m), representing numbers in decimal, hexadecimal and binary, and a basic understanding of the root words and prefixes of the metric system.
Subscribe to:
Posts (Atom)
