EQ: How do I convert between standard form and scientific notation? HW: Practice Sheet Bellwork: Simplify each expression 1. (5x 3 ) 4 2. 5(x 3 ) 4 3. 5(x 3 ) 4 20x 8 Simplify and leave in standard form 4. -7 3 5. (-12) 2
EQ: how do I operate in scientific notation? HW: Have a good weekend. Bellwork Quiz: Simplify each expression and leave in exponential form. 1. (3x 2 y 4 ) 3 2. 5(a 4 b 3 ) 5 15a 4 b 2 3. 5g -2 h 3 Simplify each expression and leave in standard form. 4. (-5) 3 5. -2 4
EQ: HOw do I operate in scientific notation? HW: Practice Sheet Bellwork: 1. (5.43 x 10 11 ) x (2.1 x 10 2 ) 2. (3.4 x 10 3 ) x (4.5 x 10 2 ) 3. (9.1 x 10 12 ) x (5.6 x 10 5 ) 4. (3.15 x 10 11 ) (2.1 x 10 5 ) 5. (5.27 x 10 11 ) (2.1 x 10 9 )
Scientific notation is nothing more than a fancy way that scientists and mathematicians represent very large and very small numbers. Can you think of some situations that would call for very large or very small numbers? Consider the following situations: The distance from the earth to the sun in centimeters The total number of cells in a square inch of human tissue The diameter of a grain of sand These are all either very large or very small numbers, and if we were doing a research study on one of the above topics, we wouldn't want to have to write that number over and over again. Thus, scientific notation is really just something we came up with because we are lazy. When you become a famous mathematician, you can be lazy too!
Before we can start writing in scientific notation, though,we have to go back and review what we know about the decimal number system. As its name implies, the decimal number system is just a number system that has 10 as its base. Think back to elementary school. Did your teacher ever use base ten blocks? Maybe when you were learning a out place value? or borrowing? *Note: yes there are number systems out there that are NOT base 10, but those are better left for another class. So let's put what we know about the decimal number system together with what we know about exponents. 10 1 or just 10 is our base What is 10 2? 100 What is 10 3? 1000 What pattern do you notice?
If you said we added a zero every time, you're not wrong, but there is a better way to put it. 10 1 = 10 10 2 = 100 10 3 = 1000 10 4 = 10000 What we are actually doing is adding a decimal place each time we raise the power of 10. The same is even true if we use negative exponents: 10-1 = 1 10 = 0.10 1 10-2 = 100 = 0.01 1 10-3 = 1000 = 0.001 So what? How does this help us write very large and very small numbers?
Consider this: What happens when you multiply any number by 10? 5 x 10 = 50 We just move the decimal 1 place to the right! What happens when i multiply any number by 10 2 (or 100)? 5 x 100 = 500 We just move the decimal 2 places to the right. let's try just one more. what happens when I multiply any number by 10 3 (or 1000)? 5 x 1000 = 5000 Exactly! I just move the decimal 3 places to the right. So the pattern is that whenever I multiply by a power of 10, I just use the exponent to tell me how many places to move my decimal. Do you think the same is true for negative exponents?
Of course it is, we just move the decimal in the other direction! 5 x 10-1 = 5 x 1 = 0.5 10 5 x 10-2 = 5 x 1 = 0.05 100 And that is the basis for scientific notation! Now the fundamentals: True scientific notation consists of 2 parts: 1. A number between 1 and 10 and 2. A power of 10 Note: We can write large or small numbers as powers of 10, but true scientific notation has a leading digit between 1 and 10. 3.21 x 10 7 Good Scientific Notation 32.1 x 10 6 Bad Scientific Notation
To convert from standard form to scientific notation: Step 1: Place the decimal point so that there is one non-zero digit to the left of the decimal. 145,000,000 1.45 Step 2: Count the number of places the decimal has been moved. 145,000,000 1.45 In this case, the decimal was moved 8 places. Step 3: Rewrite as a product of a power of 10. If the original number was greater than 1, the exponent is positive. If the original was less than 1. the exponent is negative. 1.45 x 10 8 Yes, it is that easy.
Let's try another one together. Convert the following number to scientific notation: 0.0000000000054 Step 1: Place the decimal point such that there is one non-zero digit to the left of the decimal. 5.4 Step 2: Count the number of places we moved the decimal. 0.0000000000054 5.4 We moved the decimal point 12 places. Step 3: The original number was less than one, so my exponent will be negative. 5.4 x 10-12
You try: Convert each of the following numbers to scientific notation. 1. 2,340,000,000 2. 0.00000321 3. 5,400 4. 0.000918 5. 15,000,000,000,000
To convert from scientific notation to standard form: 1. Move the decimal point to the right for positive exponents. 2. Move the decimal point to the left for negative exponents. Example: 3.41 x 10 6 3,410,000 *Note: I am NOT counting zeros! I am counting decimal places!!!!! Look at another one: 2.179 x 10-4 0.0002179 Easy enough, right?
You try some: 1. 5.45 x 10 9 2. 4.7 x 10-9 3. 2.114 x 10 12 4. 3.12 x 10-4 5. 7.8 x 10 8
Like everything else in math, we also have to be able to operate using scientific notation. What if I have two numbers in scientific notation that I want to multiply? (4.5 x 10 5 ) x (2.2 x 10 3 ) I could convert them to standard form and then multiply the traditional way: 450,000 x 2,200 = 990,000,000 And then convert back to scientific notation: 990,000,000 = 9.9 x 10 8 But that's a lot of converting. There must be an easier way. We are lazy mathematicians after all.
let's look at the problem again: (4.5 x 10 5 ) x (2.2 x 10 3 ) Are those parentheses really necessary? The commutative property of multiplication says that I can multiply terms in any order I want and still get the same answer. So it doesn't really matter how I group the terms in the problem above. Let's look at it without the parentheses: 4.5 x 10 5 x 2.2 x 10 3 Now why don't we move things around a little bit: Have I changed the problem? So... 4.5 x 2.2 x 10 5 x 10 3 4.5 x 2.2 = 9.9 10 5 x 10 3 = 10 8 (4.5 x 10 5 ) x (2.2 x 10 3 ) = 9.9 x 10 8
So what's the algorithm? Step 1: Multiply the decimal terms. Step 2. Multiply the exponential terms. Step 3: Put the two parts together. Let's look at another one: Using our algorithm, we get: (5.7 x 10 8 ) x (3.1 x 10 4 ) (5.7 x 10 8 ) x (3.1 x 10 4 ) = 17.67 x 10 12 Done, right? Not exactly. Go back to our definition of scientific notation. In order for a number to be in true scientific notation, the leading decimal has to be a number between 1 and 10. So we have one more step to do: 17.67 x 10 12 = 1.767 x 10 13
What about division? Well, since division is really just a special type of multiplication, does it make sense that division would work the same way? Step 1: Divide the decimal terms Step 2: Divide the exponential terms Step 3: Put the two parts together. Example: (9.9 x 10 7 ) (3.6 x 10 4 ) 9.9 3.6 = 2.75 10 7 10 4 = 10 3 So... (9.9 x 10 7 ) (3.6 x 10 4 ) = 2.75 x 10 3
Try another one: First we divide the decimals: (1.38 x 10 8 ) (4.6 x 10 3 ) 1.38 4.6 = 0.30 Then we divide the exponential expressions: Then we put them together again: 10 8 10 3 = 10 5 (1.38 x 10 8 ) (4.6 x 10 3 ) = 0.30 x 10 5 But that's not official scientific notation is it? 0.30 x 10 5 = 3.0 x 10 4 Now we have a solution in true scientific notation
You try some: 1. (4.3 x 10 4 ) x (2.2 x 10 3 ) 2. (6.9 x 10 8 ) x (1.3 x 10 7 ) 3. (2.9 x 10 3 ) x (4.1 x 10 9 ) 9. (2.18 x 10 9 ) (5.2 x 10 6 ) 4. (5.4 x 10 3 ) x (3.1 x 10 2 ) 5. (3.9 x 10 12 ) x (4.1 x 10 3 ) 10. (1.14 x 10 5 ) (5.2 x 10 2 ) 6. (3.4 x 10 7 ) (1.5 x 10 3 ) 7. (7.2 x 10 13 ) (2.1 x 10 8 ) Use hundredths as your level of precision for decimals that seem to go on for a while. 8. (8.7 x 10 7 ) (1.6 x 10 3 )
So what about adding and subtracting? Well, just like exponential addition, addition in scientific notation has some very strict rules it has to follow. First, we have to look and see if the power of 10 is the same. Consider the following expression: (2.34 x 10 5 ) + (5.41 x 10 5 ) Because both numbers have the same power of 10, I can just add the decimal term and leave the power of 010 the same: (2.34 x 10 5 ) + (5.41 x 10 5 ) = 7.75 x 10 5 We can see why this works if we convert to standard form: 234,000 + 541,000 775,000 Since the power of 10 just moves the decimal place, numbers with the same power of 10 can be added without changing anything.
But what if my numbers don't have the same power of 10? (1.34 x 10 4 ) + (4.45 x 10 5 ) There are two things I can do here. The first is to convert each number to standard form and then add: 13,400 + 445,000 = 458,400 Then I convert back to scientific notation: 458,400 = 4.584 x 10 5 But what if the conversion is not so simple. Consider the following: (1.35 x 10 13 ) + (2.47 x 10 12 ) Well, I can just rewrite each number so that it has the same power of 10 and then follow the rule from the previous slide: (1.35 x 10 13 ) + (2.47 x 10 12 ) = (13.5 x 10 12 ) + (2.47 x 10 12 ) (13.5 x 10 12 ) + (2.47 x 10 12 ) = 15.97 x 10 12 which is not good scientific notation, so we convert it to: 1.597 x 10 13
And subtraction works the same way: So... (5.4 x 10 9 ) - (2.1 x 10 9 ) 5.4-2.1 = 3.3 (5.4 x 10 9 ) - (2.1 x 10 9 ) = 3.3 x 10 9 Not too bad, right?
1. (4.3 x 10 4 ) + (2.2 x 10 4 ) 2. (6.9 x 10 8 ) + (1.3 x 10 7 ) 3. (2.9 x 10 12 ) + (4.1 x 10 12 ) 4. (5.4 x 10 5 ) + (3.1 x 10 3 ) 5. (3.9 x 10 15 ) + (4.1 x 10 14 ) 6. (3.4 x 10 7 ) - (1.5 x 10 7 ) 7. (7.2 x 10 13 ) - (2.1 x 10 12 ) 8. (8.7 x 10 7 ) - (1.6 x 10 7 ) 9. (8.7 x 10 7 ) - (1.6 x 10 6 ) 10. (8.7 x 10 15 ) - (1.9 x 10 15 )