Scientific Notation

Purpose of Scientific Notation Scientists are often confronted with numbers that look like this: 2,000,000,000,000,000,000,000,000,000,000 kg Can you guess what weighs this much?

Can you match these BIG objects to their weights? The Great Pyramid at Giza The Earth 300,000,000,000 kg Blue Whale - largest animal on earth 2,000,000,000,000,000,000,000,000,000,000 kg 600,000,000 kg The Sun 60,000,000,000,000,000,000,000,000 kg Total Human Population 180,000 kg

Can you match these BIG objects to their weights? The Great Pyramid at Giza The Earth 600,000,000 kg Blue Whale – largest animal on earth Click object to reveal answer 60,000,000,000,000,000,000,000,000 kg 180,000 kg The Sun Total Human Population 2,000,000,000,000,000,000,000,000,000,000 kg 300,000,000,000 kg

Can you match these small objects to their weights? grain of sand 0.00015 kg molecule 0.000000000000000000000000030 kg steam 0.00000000035 kg

Click to reveal answers. grain of sand 0.00000000035 kg molecule 0.000000000000000000000000030 kg steam 0.00015 kg

Scientific Notation The examples were written in "standard form", the form we normally use. But the standard form is difficult when a number is HUGE or tiny, if it has a lot of zeros. Scientists have come up with a more convenient method to write very LARGE and very small numbers. Writing numbers in scientific notation doesn't change the value of the number.

Scientific Notation Scientific Notation uses Powers of 10 to write big or small numbers more conveniently. Using scientific notation requires us to use the rules of exponents we learned earlier. While we developed those rules for all bases, scientific notation only uses base 10.

Writing Large Numbers in Scientific Notation

Scientific Notation Here are some different ways of writing 6,500. 6,500 = 6.5 thousand 6.5 thousand = 6.5 x 1,000 6.5 x 1,000 = 6.5 x 103 which means that 6,500 = 6.5 x 103 6,500 is standard form of the number and 6.5 x 103 is scientific notation These are two ways of writing the same number.

Scientific Notation 6.5 x 103 isn't a lot more convenient than 6,500. But let's do the same thing with 7,400,000,000 which is equal to 7.4 billion which is 7.4 x 1,000,000,000 which is 7.4 x 109 Besides being shorter than 7,400,000,000, its a lot easier to keep track of the zeros in scientific notation. And we'll see that the math gets a lot easier as well.

Scientific Notation Scientific notation expresses numbers as the product of: a coefficient and 10 raised to some power. 3.78 x 106 The coefficient is always greater than or equal to one, and less than 10. In this case, the number 3,780,000 is expressed in scientific notation.

Express 870,000 in scientific notation 1. Write the number without the comma. Place the after the first number. Write “ X 10 “ after the number. 4. Count how many places you had to move the decimal point. This becomes the exponent of 10. 5. Erase the zeros at the end of the number. 870000 8 70000 x 10 . 870000 x 10 . 1 2 3 4 5 8.7 x 105

Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1

2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 1023

5 Which is the correct coefficient of 147,000 when it is written in scientific notation? A 147 B 14.7 C 1.47 D .147 Answer: C

6 Which is the correct coefficient of 23,400,000 when it is written in scientific notation? A .234 B 2.34 C 234. D 23.4 Answer: B

7 How many places do you need to move the decimal point to change 190,000 to 1.9? A 3 B 4 C 5 D 6 Answer: C

8 How many places do you need to move the decimal point to change 765,200,000,000 to 7.652? A 11 B 10 C 9 D 8 Answer: A

Which of the following is 345,000,000 in scientific notation? 9 Which of the following is 345,000,000 in scientific notation? A 3.45 x 108 B 3.45 x 106 C 345 x 106 D .345 x 109 Answer: A

Which of these are not numbers in scientific notation? 10 Which of these are not numbers in scientific notation? A .34 x 108 B 7.2 x 103 C 8.9 x 104 D 2.2 x 10-1 E 11.4 x 1012 Answer: D F .41 x 103 G 5.65 x 104 H 10.0 x 103

Which of the following is 12,300,000 in scientific notation? 11 Which of the following is 12,300,000 in scientific notation? A .123 x 108 B 1.23 x 105 C 123 x 105 D 1.23 x 107 Answer: D

The mass of the solar system 300,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 kg (How do you even say that number?)

Writing Small Numbers in Scientific Notation

Express 0.0043 in scientific notation 1. Write the number without the decimal point. 2. Place the decimal point after the first digit. 3. Write “ X 10 “ after the number. Count how many places you had to move the decimal point. Change to negative and it becomes the exponent of 10. Or you can just count the 0’s, add one and change to negative. 5. Erase the zeros in front of the left-most non-zero digit. 004 3 x 10 . ? 004 3 x 10 . 1 2 3 ? 4.3 x 10-3

1) Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10-8

12 Which is the correct decimal placement to convert 0.000832 to scientific notation? A 832 B 83.2 C .832 D 8.32 Answer: D

13 Which is the correct decimal placement to convert 0.000000376 to scientific notation? A 3.76 B 0.376 C 376. D 37.6 Answer: A

14 How many times do you need to move the decimal point to change 0.00658 to 6.58? A 2 B 3 C 4 D 5 Answer: B

15 How many times do you need to move the decimal point to change 0.000003242 to 3.242? A 5 B 6 C 7 D 8 Answer: B

Write 0.00278 in scientific notation. 16 Write 0.00278 in scientific notation. A 27.8 x 10-4 B 2.78 x 103 C 2.78 x 10-3 D 278 x 10-3 Answer: C

17 Which of these is the only number larger than 1 in scientific notation? A .34 x 10-8 B 7.2 x 10-3 C 8.9 x 104 D 2.2 x 10-1 Answer: C E 1.14 x 10-12 F 4.1 x 10-3 G 5.65 x 10-4 H 10.0 x 10-3

Write 0.000847 in scientific notation. 18 Write 0.000847 in scientific notation. A 8.47 x 104 B 847 x 10-4 C 8.47 x 10-4 D 84.7 x 10-5 Answer: C

Converting to Standard Form

Express 3.5 x 104 in standard form 1. Write the coefficient. 2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative. 3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. 4. Drop unnecessary zeros and add comma, as necessary. 3.50000 35000.0 35,000

Express 3.5 x 10-4 in standard form 1. Write the coefficient. 2. Add a number of zeros equal to the exponent: to the right for positive exponents and to the left for negative. 3. Move the decimal the number of places indicated by the exponent: to the right for positive exponents and to the left for negative. 4. HINT – for negative exponents, you will have the same number of zeros as the exponent – only there is a decimal point after the first. 00003.5 0.00035

19 How many times do you need to move the decimal and which direction to change 7.41 x 10-6 into standard form? A 6 to the right B 6 to the left C 7 to the right Answer: B D 7 to the left

20 How many times do you need to move the decimal and which direction to change 4.5 x 1010 into standard form? A 10 to the right B 10 to the left C 11 to the right Answer: A D 11 to the left

Write 6.46 x 104 in standard form. 21 Write 6.46 x 104 in standard form. A 646,000 B 0.00000646 C 64,600 D 0.0000646 Answer: C

Write 3.4 x 103 in standard form. 22 Write 3.4 x 103 in standard form. A 3,400 B 340 C 34,000 D 0.0034 Answer: A

Write 6.46 x 10-5 in standard form. 23 Write 6.46 x 10-5 in standard form. A 646,000 B 0.00000646 C 0.00646 D 0.0000646 Answer: D Note – 5 zeros, 4 after the decimal point.

Write 1.25 x 10-4 in standard form. 24 Write 1.25 x 10-4 in standard form. A 125 B 0.000125 C 0.00000125 D 4.125 Answer: B Note – 4 zeros, 3 after the decimal point.

Write 4.56 x 10-2 in standard form. 25 Write 4.56 x 10-2 in standard form. A 456 B 4560 C 0.00456 D 0.0456 Answer: D Note – 2 zeros, 1 after the decimal point.

Write 1.01 x 109 in standard form. 26 Write 1.01 x 109 in standard form. A 101,000,000,000 B 1,010,000,000 C 0.00000000101 D 0.000000101 Answer: B