➕ Algebra 1 · Exponents and Polynomials
Scientific Notation: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
Idle-game scoreboards
In some idle and clicker games, the score grows so large that the game stops showing every digit and switches to something like 3.4e8. That is the game’s way of printing scientific notation.
How it connects: The display 3.4e8 means 3.4 × 108, which is 340,000,000. The e stands for “times ten to the power,” and it is not the special number 2.718... from later math courses. Limit: a game or calculator display may round the front number, so 3.4e8 can be only an approximation of the exact score.
The range of hearing
People with typical hearing can hear sounds from about 20 vibrations per second (hertz) up to about 20,000 hertz. The exact range varies from person to person and shrinks with age. Musical notes sit inside this range: the lowest key on a standard 88-key piano vibrates about 27.5 times per second.
How it connects: In scientific notation, that range is about 2 × 101 to 2 × 104 hertz. Dividing gives (2 × 104) ÷ (2 × 101) = 1 × 103, so the highest sounds people hear vibrate about 1,000 times as fast as the lowest. Limit: the exponents differ by 3, and that gave exactly 1,000 only because both front numbers are 2. When the front numbers differ, you must divide them too.
Alice in Wonderland (the book and its films)
In Lewis Carroll’s Alice’s Adventures in Wonderland, and the films based on it, Alice drinks from a bottle labeled DRINK ME and shrinks, and later eats something labeled EAT ME and grows. Pretend that each sip of the shrinking drink makes her one tenth as tall as before. After 3 sips she is 10−3 times her starting height.
How it connects: Shrinking gives a negative exponent, and growing would give a positive one. A negative exponent here means “a fraction of the original size,” not a negative size. An Alice who starts at 1.6 meters tall would end up 1.6 × 10−3 meters tall, which is 1.6 millimeters. Limit: the one-tenth-per-sip rule is made up for this picture. The story has no such rule.
Get on the same page
To “get on the same page” means to make sure everyone shares the same understanding before moving on.
How it connects: Before you add or subtract numbers in scientific notation, get them on the same page: the same power of 10. Then 4 × 102 + 7 × 101 becomes 4 × 102 + 0.7 × 102 = 4.7 × 102, which matches 400 + 70 = 470. Limit: this is only for adding and subtracting. When you multiply or divide, the powers do not have to match.
Marathon versus sprint
A 100-meter dash is 1 × 102 meters long. A marathon is 42,195 meters, which is about 4.2 × 104 meters.
How it connects: To see how many times as long the marathon is, divide: (4.2 × 104) ÷ (1 × 102) = 4.2 × 102, which is about 420 times as long. Limit: the exponents differ by 2, but the answer is not just 100 times, because the front numbers (4.2 and 1) matter too. The exponent difference only gives the power of 10 in the answer.
The ladder of sizes
A typical atom is about 1 × 10−10 meters across, and a human hair is about 1 × 10−4 meters wide.
How it connects: How many times as wide is the hair? Divide: (1 × 10−4) ÷ (1 × 10−10) = 10−4−(−10) = 106, so the hair is about a million times as wide as a typical atom. Notice that −4 − (−10) is +6, not −14. Limit: both sizes are rounded to one digit, and atoms vary in size, so “about a million” is a rough comparison, not an exact count.
Reading a nutrition label
Suppose a snack label says it has 450 milligrams of sodium. A milligram is 10−3 grams.
How it connects: So 450 milligrams is 450 × 10−3 grams, which is 0.45 grams, or 4.5 × 10−1 grams in scientific notation. Changing units changes the numbers we write but not the amount. Limit: 450 × 10−3 is not in scientific notation, because 450 is not less than 10. Also, convert to one unit before you add or subtract amounts, and check that the units fit together before you multiply or divide.
The step counter
Why did the decimal point wear a step counter? In scientific notation, the exponent counts every step it takes.
How it connects: Slide the decimal point of 0.00083 right 4 places to get 8.3, and you get 8.3 × 10−4. The size of the exponent counts the steps, and the sign records the direction: right for a number below 1 gives a negative exponent, and left for a number of 10 or more gives a positive one. Limit: it counts the steps of the point, not the zeros.
Memory help
MOVE
To convert a number to scientific notation: Move the decimal point until exactly one nonzero digit is to its left. Observe how many places it moved: that count is the size of the exponent. Verify the direction: sliding left (a number of 10 or more) gives a positive exponent, and sliding right (a number between 0 and 1) gives a negative exponent. Express the answer as the front number times the power of 10.
Where the exponent sits
Ten or more? The exponent is above zero. Under one? It dips below. From one up to ten, it sits at zero, and the slides you count show how far you go. (This is for positive numbers.)
Match, Add, Fix
For adding or subtracting numbers in scientific notation: Match the powers of 10 (rewrite one number), Add or subtract the front numbers and keep the power, then Fix the front number if it left the range from 1 up to (but not including) 10. Multiplying and dividing skip the Match step, because their powers do not have to match. They use the level-down rule from Laws of Exponents: multiplying adds exponents and dividing subtracts them.
Draw the hops
Write the number and draw one small arc for every place the decimal point slides, from where it starts to where it stops. Number the arcs 1, 2, 3 and so on. The last number is the size of the exponent, and the direction of the arcs gives the sign: left is positive, right is negative. Counting arcs keeps you from counting zeros by mistake.
The tidy-up crew
After you multiply or divide, picture a tidy-up crew checking the front number. If it is 10 or more, the crew slides its point left and tells the exponent to go up. If it is less than 1, the crew slides the point right and tells the exponent to go down. Each slide changes the front number by a factor of 10, and the exponent changes to make up for it, so the value never changes.
Count the slides, not the zeros
The number 1,700,000 has five zeros, but it is 1.7 × 106, not 1.7 × 105. The exponent is the number of places the decimal point slides, not the number of zeros. For a number with a decimal like 0.009, count the slides: past 0, 0 and 9 is 3 slides.