Scientific Notation Quest

Welcome Brave Adventurer! For many years, you have studied at the Academy of Fantastical Engineering. You've always worked hard, but recently, you've made a discovery that you think might change the world. Your teachers and fellow students disagree. They think it's too simple, but you are sure that this is much more important than it seems. You've got a plan, though. A three-part plan. First, you'll make sure you understand your discovery. Then, you'll go out into the world and collect practical applications that you can use as evidence to prove your theory. Lastly, you'll bring your discovery and your evidence to the head of your academy. Or maybe you'll skip all that and dive right into convincing the head of your academy . . . [[Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Bring your discovery to the head of your academy]]

You sit down on a grassy hill and pull out a piece of parchment to consider your discovery. Your pixie flutters over your shoulder. It all started because you were thinking about place value. Early on, you learned that numbers like 357 meant you had 3 hundreds, 5 tens, and 7 ones. Then, you started noticing other patterns. All having to do with the what happened when you multiplied numbers by 10, 100, 1,000, etc. You noticed that: 5 x 10 = 50 6 x 10 = 60 48 x 10 = 480 Sure, it was kind of like you added a zero on the end if you multiplied by 10, but really what was happening meant more than that. Everything in the ones place moved to the tens place. (Because 1 x 10 = 10.) Everything in the tens place moved to the hundreds place. Basically, multiplying by 10 simply shifted all the numbers to a higher place value. It was better to think of that rather than "just adding a zero" you realized, because sometimes you had weird questions like this: What's 5.8 x 10? [[I can figure that out!]] [[Ask your pixie]]

You excitedly request an audience with the head of your academy: Helen the Builder, Lead Architect and Head of the Society of Fantastical Engineering. Your request is denied. [[Show up at her offices anyway and try to talk to her]] [[Work on your theory a bit more->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

You set off into the world, travelling for many weeks. You hike through fields of butterflies, climb to the top of lonely peaks, and get lost in vast stone cities. One day, you come to a small harbor where people seem friendly and everyone is cheerfully going about their business. You notice some dwarves who look worried, though. One is a dwarf with a red captain's hat. He's counting out gold coins on the wharf. [[Approach the Dwarf on the Dock]]

Correct! 5.8 x 10 = 58 Whenever you multiply by 10, you shift the decimal point over 1, which moves all the digits to a bigger place value. How about a trickier one? What's 64.87 x 10? [[I can try that!]]

Not quite! for 5.8 x 10 we move the decimal point one place to the right, which gives us 58. What about 7.1 x 10? [[Let me see...]]

(set: $answer to (prompt: "Type your answer to 5.8 × 10 here:" , "")) (if: $answer is "58")[ (goto: "5.8x10 Correct") ] (else:)[ (goto: "5.8x10 Try Again") ]

"How do I multiply 5.8 by 10 again?" you ask your pixie. "Good question," she says. "We could of course do long multiplicaiton by hand. But, as you noticed before, multiplying by 10 turns every 1 to a 10, and every 10 to a 100. For example like this: 7 x 10 = 70 (7 ones became 7 tens) 98 x 10 = 980 (8 ones became 8 tens, and 9 tens became 9 hundreds) "So, even if there's a decimal place, everything still just shifts over. For example, like this: 2.3 x 10 = 23 7.4 x 10 = 74 [[I think I see now. Let's try 5.8 x 10 now... ->I can figure that out!]]

(set: $answer to (prompt: "Type your answer to 64.87 × 10 here:" , "")) (if: $answer is "648.7")[ (goto: "64.87x10 Correct") ] (else:)[ (goto: "64.87x10 Try Again") ]

(set: $answer to (prompt: "Type your answer to 7.1 × 10 here:" , "")) (if: $answer is "71")[ (goto: "7.1x10 Correct") ] (else:)[ (goto: "7.1x10 Try Again") ]

Correct! 7.1 x 10 = 71 Whenever you multiply by 10, you shift the decimal point over 1, which moves all the digits to a bigger place value. How about a trickier one? What's 64.87 x 10? [[I can try that!]]

Not quite! for 7.1 x 10 we move the decimal point one place to the right, which gives us 71. What about 9.5 x 10? [[95]]

How about a trickier one? What's 64.87 x 10? [[I can try that!]]

Correct! 64.87 x 10 = 648.7 Whenever you multiply by 10, you shift the decimal place over 1, which moves all the digits to a bigger place value. What do you think would happen if we multplied by 100 instead? [[The decimal would move over two spaces]] [[I'm not sure...]]

Not quite! for 64.87 x 10 we move the decimal point one place to the right, which gives us 648.7 What about 439.21 x 10? [[4,392.1]]

Very true! The decimal would move over two spaces. Any guesses on what 5.3 x 100 would be? [[Sure!]] [[Hm that seems hard.]]

Let's look at some examples! 5 x 100 = 500 6.78 x 100 = 678 (which makes sense because 6 x 100 is 600, so it makes sense that the .78 just comes along with the 6) 85 x 100 = 8500 How would you describe the trick for multiplying by 100? [[Move the decimal point two places to the right (or add two zeroes, if there is no decimal)->The decimal would move over two spaces]]

Correct! 439.21 x 10 =4,392.1 Whenever you multiply by 10, you shift the decimal place over 1, which moves all the digits to a bigger place value. What do you think would happen if we multplied by 100 instead? [[The decimal would move over two spaces]] [[I'm not sure...]]

(set: $answer to (prompt: "Type your answer to 5.3 x 100 here:" , "")) (if: $answer is "530")[ (goto: "5.3x100 Correct") ] (else:)[ (goto: "5.3x100 Try Again") ]

It is hard! If we try to move the decimal two spots, there's nowhere for it to go the second time! But, remember that 5.3 is the same as 5.30, or 5.3000000, we just don't write all those zeroes because we don't need to. But, if we were trying to mulitply 8.5 x 100, we could first think of it as 8.50 x 100. We'd move the decimal right twice to get 850! [[Ah. I see. I can try 5.3 x 100 now. ->Sure!]] [[Can I see some more examples?]]

Sure thing! Remember there are hidden zeroes there that we don't usually write but that we can use if we need them: 7.1 x 100 Becomes 7.10 x 100 Now that we included that zero on the end, it's easier to move the decimal place right twice. Moving it right twice gives us 710 89.3 x 100 is 8,930 65 x 100 is 6500 [[Okay. I can try 5.3 x 100 now. ->Sure!]]

Correct! 5.3 x 100 = 530 This is your big discovery. It might seem simple, but you're very sure it will change the world. Multiplying by 10 means moving the decimal right once. Multiplying by 100 means moving the decimal right twice. Mulitplying by 1,000 means moving the decimal right three times. What about multiplying by 100,000? How many times would you move the decimal place to the right in that case? [[5 times]] [[6 times]]

"Not quite," your pixie says. "Remember that multiplying by 100 means moving the decimal right twice." [[Ah. I see. I can try 5.3 x 100 now. ->Sure!]] [[Can I see some more examples?]]

Correct! You can always count up the zeroes. 100,000 has 5 zeroes, so it's the same as 5 tens multiplied together. Each time you multiply by a ten, you'd move the decimal point once. So, 5 tens means move the decimal point 5 times. You've got it now. You've reviewed everything about your world-changing idea. Now if you could only prove how this can change the world! [[Bring your discovery to the head of your academy]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

"This one is tricky!" your pixie says. "There are six numbers here, so it feels like it should be six. But remember this pattern: x10 = move the decimal once 100 is the same as 10 x 10, so multiplying by 100 is the same as multiplying by 10 twice x 100 = move the decimal twice 1,000 is the same as 10 x 10 x 10, so x 1,000 = move the decimal three times What do you notice about the number of zeroes compared to the number of times you move the decimal? [[The number of zeroes is the same as the number of times you move the decimal]]

Correct! What would that mean about multiplying by 10,000,000? How many times would we move the decimal over? [[6->8]] [[7]] [[8]]

Correct! Multiplying by 10,000,000 means moving the decimal right 7 times. You've got it now. You've reviewed everything about your world-changing idea. Now if you could only prove how this can change the world! [[Bring your discovery to the head of your academy]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

Not quite! For 10,000,000, there are 7 zeroes, so we would move the decimal over 7 times. [[Ah, okay, got it.]]

You've got it now. You've reviewed everything about your world-changing idea. Now if you could only prove how this can change the world! [[Bring your discovery to the head of your academy]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

=><= ''Scientific Notation Quest'' Written by Sarah Allen Illustrated by Asanee Srikijvilaikul [[Start]]

You wait outside the beautiful and intimidating office of Helen the Builder, most skilled and famous of all the fantastical engineers, until you see her return. A large sign on the door says "No Admittance Without Appointment - Please Knock." [[Knock on the front door]] [[Walk right in]] [[Sneak around the back]]

You knock on the front door. There's a long wait, but you finally hear someone coming. A man with his arms full of rolled-up building plans opens the door, looking busy. "Yes?" [["Hi. I'm sorry to bother you. Can I please speak to the head engineer?"]] [["There's an emergency! I need to speak to the head engineer immediately!"]]

You confidently open the door and stride right in. You're in a large entryway. There are long halls to the left and right. Light glows from a room just ahead, and you can hear voices. Is that the architect? Down the hall to the right, there's a man carrying a bunch of building plans. He stops short with a gasp, dropping all the plans. [[Smile and help him with the plans]] [[Run towards the architect!]]

You clamber over the rocks and sneak around the back. There's a small window about six feet off the ground, and a single, plain door. [[Try the door]] [[Try to climb up to the window]]

"What's this about?" [["I've made a really important mathematical discovery, and I think it would help a lot. Can I please speak to her?"]]

He blinks a few times, readjusting the papers. "What kind of an emergency?" [[A fire]] [[Bandits]] [[Umm . . . an architectual emergency?"]]

He blinks at you. One of the rolled-up plans slips out of his grasp. Then the rest of them follow. He scrambles to pick them up. "You need to file a request to speak with her." [["I did that."]] [[Help him pick up the papers.]] [[Help him pick up the papers. "I did that, but it was denied. But I really think she needs to hear this.]]

"Well, then, you should expect an answer in ten to eleven days, excluding holidays," he says, grumbling. He finishes picking up his papers, gives you a polite nod, and shuts the door. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

"Thank you," he says when the papers are tucked firmly back into his arms. "You're welcome. Is there any way I could please speak to the head engineer?" "Of course." "Oh thank you!" "She's always happy to speak to people. Who file requests. Please file a request." He closes the door, and your heart crashes like a poorly designed building. Ugh. You'd really thought you'd made it in that time. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Sneak around the back]]

"Thank you," he says when the papers are tucked firmly back into his arms. He gives you a long look. You try to look as earnest and serious as possible. Finally he glances back over his shoulder. "Well, she does happen to have just a quick minute free right now. But only a minute." [["Thank you so much! I'll be quick!"]]

"Goodness, well, quick, get the fire brigade! That's not an architectual problem!" He stares over your shoulder, scanning the horizon, clearly looking for a fire. "Where is it?" he asks. You point vaguely behind you. He drops his papers, slams the door, locks it, and runs off in the direction you're pointing. You're left standing alone, with only the sound of the lapping waves and the call of seagulls. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Sneak around the back]]

He blanches, dropping the papers. "Well then call the guards!" Shaking, he slams the door shut again. You hear a heavy bolt sliding into place. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Sneak around the back]]

His eyes narrow. "Any building designed by the head fantastical engineer would never result in any kind of emergency. We pride ourselves on careful design. Thinking through every eventuality. That is how emergencies are avoided." He scans the road behind you suspiciously, then slowly shuts the door. You hear a bolt sliding into place. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Sneak around the back]]

Helen the Builder is tall even though she's sitting at a long wooden table with complex building plans spread out all over it. She looks like she's in her late sixties, and an eagle sits on her shoulder, its eyes closed. "Well, what's your idea?" she asks. "Well, you see, I've discovered a trick about multiplying numbers by 10, 100, 1,000, etc." "Multiplying by powers of ten?" Her eyebrows raise. "What's the trick?" "Well, all you have to do is move the decimal place right." "Can you give me an example?" [[You realize you need to think about your discover a bit more. "Actually. Nevermind," you say. "I'd better go." ->Think about your discovery]] [["Of course. For example, 8.19 x 1,000"]]

You grasp the handle and pull. It's locked. [[Pull harder]] [[Try to climb up to the window]]

You skillfully climb the building. Luckily it's got a lot of decorative bits that make good handholds and footholds. You reach the window in a flash. It's open! You start to climb inside, but you're only halfway in when a hand grabs your foot. "Hey! What do you think you're doing?" a man shouts. Two guards, one standing on the other's shoulders to reach you, stand below you. They haul you down from the window. You try to explain that you were just needing to talk to the head engineer, but they won't listen. You're still trying to explain your theories as they march you out of town. [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Maybe it's a better idea to think more about your theories for a bit.->Think about your discovery]]

You yank hard on the door, but it's a very well-built door. You throw your whole weight into it, shaking it back and forth. "What do you think you're doing?" a voice says. Two guards come around the corner. You explain that you've discovered a very important mathematical truth and need to speak to the head engineer. They glance at each other and roll their eyes. "Man. Another one," one of them says. "Did you make an appointment?" the other says. You explain that you did, but were denied. "Well, you can always try again later," one of them says. "In the meantime, we'd like you to leave the premises." [[Agree, but sneak around the other side and look for another window]]

You find another window, a perfect, big window right at ground level! Unfortuantely, just as you start to try to open it, the guards show up. You're still trying to explain your theories as they march you out of town. [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Maybe it's a better idea to think more about your theories for a bit.->Think about your discovery]]

You move to help him with the plans, but he shrieks and runs away, shouting. He seems like an oddly jumpy person, you think. Two guards show up a moment later. "Excuse me," one says. "Do you have an appointment?" [["No, but I really need to speak to the head engineer."]] [["Yes, of course."]]

You burst into the lit central room! Long tables run the length of the room, lit by oil lamp and filled with piles of papers. At the end of the hall sits the head engineer! Helen the Builder! She glances up, with a raised eyebrow. From the other side of the room, you see two guards sprinting towards you. [["I'm sorry to bother you can I please tell you about a math thing I--aagh!"]]

"Why?" [[I've discovered a really cool mathematical thing!]] [["There's an emergency!"]]

The jumpy man returns, peering at you from behind the two guards. "We don't have any appointments scheduled today," he says. The guards frown at you. They escort you from the building, and then from the town. [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Maybe it's a better idea to think more about your theories for a bit.->Think about your discovery]]

The first guard reaches you and tackles you to the ground. All the air goes out of your lungs. He hauls you to your feet. You're still gasping for breath. Helen the Builder raises a hand. "You're the one with the mathematical theory, aren't you?" She's holding a compass with points on both ends, and she taps it thoughtfully. "Well, I suppose I have a minute. You're very committed. But, only a minute." The guards eye you suspiciously. [["Thank you so much! I'll be quick!"]]

"Please make an appointment, then," he says. [["I tried."]]

He blinks a few times, readjusting the papers. "What kind of an emergency?" [[A fire->fire2]] [[Bandits->bandits2]] [[Umm . . . an architectual emergency?"->arch3]]

"Then what in the world are you doing here?! Call the fire brigade!" They march you out of the building, and one of the guards runs off. The other stands watching you with his arms crossed. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

One of the guards crosses his arms. "And you want an engineer for that?" They glance at one another. You feel a blush heating your face. "Umm . . . I guess not." "We'll contact the rest of the guards. You can go back about your business." Frustrated, you leave. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

His eyes narrow. " Any building designed by the head fantastical engineer would never result in any kind of emergency. We pride ourselves on careful design. Thinking through every eventuality. That is how emergencies are avoided." The guards escort you out of the building. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Sneak around the back]]

"Well, then, you should expect an answer in ten to eleven days, excluding holidays," he says, grumbling. A guard shows up. They're very tall. "Is there a problem here?" He looks very menacing. "No, sorry," you say. "This person wanted to meet the head engineer. I've told them to make an appointment." The guard lifts his eyebrow at you. "Yes, sorry," you mumble. "I'll make an appointment." You leave through the front door. [[Go back to think about your discovery some more ->Think about your discovery]] [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

(set: $answer to (prompt: "Type your answer to 8.19 × 1,000 here:" , "")) (if: ($answer is "8,190") or ($answer is "8190"))[ (goto: "8.19 × 1,000 Correct") ] (else:)[ (goto: "8.19 × 1,000 Try Again") ]

She nods, impressed. "That is true. But what about division, can it be used for that?" [["Yes! For example 73.2 divided by 1,000.]] [[You realize you need to think about your discover a bit more. "Actually. Nevermind," you say. "I'd better go." ->Think about your discovery]]

"Hmm," she says. "That's not correct, though. 8.19 x 1,000 is 8,190." "Oh. Right," you say. You are certain that your theory is useful and important, and you want to be sure of convincing Helen the Builder. You decide to go think about your theories just a bit more. "Do you mind if I come back another day?" "Of course not," she says. "Just make an appointment." [[You thank her and go to work on your theories a bit more->Think about your discovery]] [[You decide to go test your theories in the world first->Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

(set: $answer to (prompt: "Type your answer to 73.2 divided by 1,000 here:" , "")) (if: ($answer is "0.0732") or ($answer is ".0732"))[ (goto: "73.2 Correct") ] (else:)[ (goto: "73.2 Try Again") ]

"Hmm," she says. "That's not correct, though. 73.2 divided by 1,000 is 0.0732." "Oh. Right," you say. You are certain that your theory is useful and important, and you want to be sure of convincing Helen the Builder. You decide to go think about your theories just a bit more. "Do you mind if I come back another day?" "Of course not," she says. "Just make an appointment." [[You thank her and go to work on your theories a bit more->Think about your discovery]] [[You decide to go test your theories in the world first->Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

"Very true!" she says. "Well. That is a handy trick. And it'll make our calculations just a bit faster. But, is that all? Is there some other more practical use?" [["I'd hoped you would have an idea how to use it."]] [["Yes! It means we can much more efficiently write very big and very small numbers."]]

"Oh," she says, sitting back in her chair. "Sometimes even with great mathematical discoveries, it can take some time to find practical applications. I don't have any ideas at the moment, but I can see that you are very smart and resourceful. I'm sure you can think of good applications. I look forward to hearing about them someday. In the meantime, I'd better get back to my work." "Of course, I understand," you say, deeply disappointed. Still, you're not going to give up. You will think more about your theories, and maybe go out into the world, searching for practical applications. Temporarily defeated, you leave Helen the Builder to her work. [[Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]] [[Think about your discovery]]

She steeples her fingers and leans forward. "I'm interested. Show me." "Of course. Take, for example, the number 93,000,000,000. If we were doing calculations with it, we would have to keep writing all those zeros. But there's a much faster way! We can write it in what I am now calling 'Scientific Notation.'" "Ah, that's fascinating. What would this very large number look like in this 'Scientific Notation'?' [["Like this!"]] [[You're so excited that she's intrigued by your idea, but you realize you want to explore your idea a bit more first. ->Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

(set: $answer to (prompt: "Type 93,000,000 in scientific notation here. Your answer should look like #.#x10^# or #.#E#" , "")) (if: ($answer is "9.3x10^7") or ($answer is "9.3E7"))[ (goto: "9.3 Correct") ] (else:)[ (goto: "9.3 Try Again") ]

"Wow! Most impressive!" Helen the Builder says. "And . . . does it work for very small numbers, too?" "It does! I'll show you!" [["I'll write 0.000068 in scientific notation."]]

"Hmmm shouldn't it have been 9.3 x 10^7, though?" Helen asks. She's very quick, understanding your new thoeries! "Oh, right, oops. Because from 9.3, you'd have to move the decimal right 7 times to make it back into 93,000,000." You are certain that your theory is useful and important, and you want to be sure of convincing Helen the Builder. You decide to go think about your theories just a bit more. "Do you mind if I come back another day?" "Of course not," she says. "Just make an appointment." [[You thank her and go to work on your theories a bit more->Think about your discovery]] [[You decide to go test your theories in the world first->Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

(set: $answer to (prompt: "Type 0.000068 in scientific notation here. Your answer should look like #.#x10^# or #.#E#" , "")) (if: ($answer is "6.8x10^-5") or ($answer is "6.8E-5"))[ (goto: "6.8 Correct") ] (else:)[ (goto: "6.8 Try Again") ]

"Hmmm shouldn't it have been 6.8 x 10^-5, though?" Helen asks. She's very quick, understanding your new thoeries! "Oh, right, oops. Because from 6.8, you'd have to move the decimal left 5 times to make it back into 0.000068." You are certain that your theory is useful and important, and you want to be sure of convincing Helen the Builder. You decide to go think about your theories just a bit more. "Do you mind if I come back another day?" "Of course not," she says. "Just make an appointment." [[You thank her and go to work on your theories a bit more->Think about your discovery]] [[You decide to go test your theories in the world first->Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

Helen the Builder stands up so suddenly that the eagle blinks its eyes open and flaps off to one of the rafters. "This is incredible!" she says. "We will save so much time, not having to write such huge numbers out over and over again!" "Exactly!" you shout, elated that she's seeing the value of your idea. "Not only that, but we'll avoid so many small mistakes." "You're right!" she shouts. "The number of times I've miscounted the zeroes in a number and then later a building collapsed . . ." she shakes her head. "Terrible." She pauses, staring off into space. "But, is it hard to go the other way? Like, if you have a number in scientific notation, is it hard to put it back into standard notation?" "Not at all!" you say. "Here, I'll show you!" [[Write 9.04 x 10^-2 in standard notation]]

(set: $answer to (prompt: "Type 9.04 x 10^-2 in standard notation here:" , "")) (if: ($answer is "0.0904") or ($answer is ".0904"))[ (goto: "9.04 Correct") ] (else:)[ (goto: "9.04 Try Again") ]

"Holy cow, you're right!" Helen shouts. The guards and the assisstant architect have all come closer to watch. Their eyes are wide and they're staring at you as if they've never seen someone so good at math before. "This is an amazing invention!" Helen the Builder says, coming over to you and placing a weathered hand on your shoulder. "Thank you for discovering this, and for bringing this to my attention." You blush, feeling very warm. It was a hard road to get here, but you believed in your discovery, and you achieved what you set out to do. Helen the Builder immediately appoints you as "Head Expert in Number Theory and Scientific Notation." She gives you a sash and a very fancy hat with several gold stars on it. From that day forward, scientific notation becomes the most widely used tool for working with very large and very small numbers. The fields of engineering and science both become more accurate and more efficient. And in the future, when you have more cool ideas and theories, everyone is eager to listen. The End [[Play Again->Start]]

"Hmmm shouldn't it have been 0.0904, though?" Helen asks. She's very quick, understanding your new thoeries! "Oh, right, oops. Because you take the 9.04 and move the decimal left twice. You have to include the extra 0 as a placeholder, which is extra tricky. You are certain that your theory is useful and important, and you want to be sure of convincing Helen the Builder. You decide to go think about your theories just a bit more. "Do you mind if I come back another day?" "Of course not," she says. "Just make an appointment." [[You thank her and go to work on your theories a bit more->Think about your discovery]] [[You decide to go test your theories in the world first->Set off into the world to find practical applications of your theory that you can use to convince everyone you're right]]

The man has a distracted, tense look, and his hair sticks up at odd angles. He glances up as you approach, startled. "Can I help you?" you ask. "Urghgghh," he groans, tugging his hair. "I'm a carpenter, and I'm ordering materials for a home I'm going to build soon. But my records are a mess." He holds out his records. You see notes and numbers everywhere: 1,000 tiles 20,000 nails It's more big numbers! [["I can help you with that!"]]

The dock creaks and sways under your feet as you approach the dwarf. "Excuse me sir," you say. "Is everything all right?" He takes off his hat, giving a bow. "Quite all right, except my ledgers are an absolute mess." "Ledgers?" "My banking books! My records!" He pulls out a leatherbound book several inches thick with pieces of paper sticking out from it at odd angles. He flips it open and you see lists and lists of numbers: records of money the dwarf has lent or borrowed. $5,000 $600 $40 $30,000 ... He points to some squiggles at the ends of the numbers. "You see, I end up having to write so many zeros that sometimes I miss one, or write too many, or they blur together. And I'll tell you, my business won't last long if I give someone $60 who's expecting $600!" A thrill runs through you. This is it. This is the moment you've been waiting for. Your mathematical discovery is the perfect solution to this. [["I think I can solve your problem!"]] [[You want to be extra certain you fully understand your theory so you can apply it, so you decide to go think about it some more first.->Think about your discovery]]

"You can?" the dwarf asks. "Oh that would be amazing. What's your idea?" [["Make people remember the amounts themselves. That way you don't have to keep records at all!"]] [["Keep track of the size of the number separately from the number itself."]] [["Only lend or borrow the exact same amount of money from everyone? Then you only have to write their names."]]

His eyes glaze over. "Ah. Hm. Right. Good idea. People can definitely be trusted to tell me exactly how much money I owe them." He gives you a polite smile, makes an excuse, and hurries off. [[Approach another dwarf->Approach the Dwarf on the Dock]]

He looks confused. "What do you mean by 'keep track of the size of the number'?" he asks. [[You start to feel like you want to review your discovery a bit more before talking any further->Think about your discovery]] [["Well, see, you could just write down the number 6 and the fact that the 6 is in the hundred's place."]]

He twirls his moustache. "Huh. I suppose that could work . . . But . . . what if people need more money than just $100 or so?" "Maybe just keep track of how many hundreds you lend each person?" He tilts his head to the side. "Not a terrible idea. Out of the box. Certainly does simplify the record-keeping. . ." A customer comes up to request some money from him, and the dwarf moves off. You feel happy to have helped, but still no closer to finding how your math discovery could help people in the real world. [[Approach another dwarf->Approach the Dwarf on the Dock]]

Excitement dawns in his face. "Ah, what a fascinating idea. So, instead of $30,000, I could write 3 and ten-thousands." He smooths his beard thoughtfully. "Yes," you say. "But you don't even need words. You can write it mathematically." $5,000 = 5 x 1,000 $600 = 6 x 100 $40 = 4 x 10 $30,000 = 3 x 10,000 "But this is even worse than before!" he says. "I'd still have to write all those zeros, but a bunch of other stuff, too." He starts to walk off. "Wait!" you shout. "There's a way to shorten those numbers with all the zeros!" You have an idea, but you're not totally sure if it'll work. You think it might have something to do with . . . exponents. [[Ask your pixie to tell you about exponents->Ask your pixie to remind you about how exponents work]] [[Tell the dwarf about exponents]]

The dwarf turns around, lifting his eyebrows. "See," you continue excitedly. "Each of the big numbers with all those zeros can be written as an exponent. Like this." You pull out a piece of paper and make some notes. 100 = 10^^2^^ His eyes widen. "Wow. So instead of writing $30,000. . ." "You could write 3 x 10^^4^^ instead!" "Wow! That is much more efficient!!" He looks absolutely delighted. You feel lighter on your feet as he pulls out a fresh book and starts rewriting all his notes using your new way of writing. You decide to call it "scientific notation" and the rule will be that you write the digits of the number starting in the one's place, like 5 or 6.2, and then the power of ten after that. You're almost there. You think you've nearly got it, but you want to gather just a bit more evidence before you head back to try to convince the head of the academy. Maybe there's just one or two more people here you could help . . . [[Look around the harbor more]]

"Are you sure?" he asks. "Would 10^^4^^ mean 10x10x10x10? And isn't that 10,000?" [["Oh. Right. I meant 10^^3^^. Which can also be written as 10^3."->"1,000 could also be written as 10^3!"]]

"Hey buddy," your pixie says, appearing at your shoulder. "How was your vacation?" you ask. "Oh fine. The family and I ran into a seven-fanged time monster, but we managed to defeat it. The trick for dealing with seven-fanged time monsters is—" "Sorry to interrupt, would you mind explaining exponents to me?" "Oh, definitely. Exponents are a real time-saver for people who do math." [["Great! How do they work?"->"How do they work?]]

"Well, let's say you were going to multiply three fives together, like 5 x 5 x 5. You could instead just say 'five to the third power' and write it like this: 5^^3^^ We call that little 3 an 'exponent', and it means there are 3 fives we're multiplying together. For another example, 4 x 4 could be written as 'four to the second power, which we might also call 'four squared', or 4^^2^^. Here's one more example: 10^^4^^ = 10 x 10 x 10 x 10. An exponent is usually written as a superscript—text placed above the line (from Latin super, which means “above,” and scriptus meaning “written”). It can be hard to type, so sometimes people will write this: 10^4 You would still read it aloud as 'ten to the fourth power' or just 'ten to the fourth.'" "It looks pretty weird!" "It does. But it's not as bad as it looks. On a keyboard, to type the ^ symbol you usually just hold shift and hit the 6 key. Can I give you a few to try for practice?" You sigh. Your pixie loves giving practice problems. You suppose it does help you remember things, though. "What would 8 x 8 x 8 x 8 x 8 x 8 be?" [["Hmm. I think I can try that . . ."]]

(set: $answer to (prompt: "Rewrite 8 x 8 x 8 x 8 x 8 x 8 using exponents. Type your answer here using the ^ for the superscript. (To type a ^ on most computers, hold shift and type the 6 key.) Your answer should look like '8^#'" , "")) (if: ($answer is "8^6"))[ (goto: "8^6 Correct") ] (else:)[ (goto: "8^6 Try Again") ]

"Correct!" Your pixie says. "What about the other way? What would 3^4 mean?" [["I can do that."]]

"Not quite," your pixie says. "How many 8s are there in 8 x 8 x 8 x 8 x 8 x 8?" "It's hard to count," you say. "They all blend together." "True," your pixie says. "Try counting it a few times, holding your finger or pencil over the nubers as you go. Try counting forwards and then backwards and make sure you get the same thing." "Okay . . . I think there are [[5]] [[6]] [[7->5]]

(set: $answer to (prompt: "Rewrite 3^4 as a bunch of 3s multiplied together with no spaces. Your answer should look like 3x3x3x... (You can use the lower case x as the multiplication symbol.)" , "")) (if: ($answer is "3x3x3x3")or ($answer is "3*3*3*3") or ($answer is "3 x 3 x 3 x 3"))[ (goto: "3^4 Correct") ] (else:)[ (goto: "3^4 Try Again") ]

"Very nice," your pixie says. "So, that's exponents!" "Wonderful," you say. "So, thinking about how this applies to my work, I know this: 100 = 10 x 10 = 10^^2^^ 1,000 = 10 x 10 x 10 = 10^^3^^ "Interesting," you say. "I've just noticed that the total number of zeros is the same as the exponent." "Ah yes! Very good point! What would that make 10,000,000?" your pixie says trickily. [["Hmmm. . ."]]

"Not quite," your pixie says. "How many 3s should you be writing multiplied together, if the exponent is 3^^4^^?" [[Three]] [[Four]]

(set: $answer to (prompt: "Rewrite 10,000,000 using exponents. Type your answer here using the ^ for the superscript. Your answer should look like '10^#'" , "")) (if: ($answer is "10^7"))[ (goto: "10^7 Correct") ] (else:)[ (goto: "10^7 Try Again") ]

"Perfect!" your pixie says. "Any more questions?" "I think I've got it," you say. "Good, because my sister just ran into a 5-eyeballed blob-claw and she needs some help." "Good luck!" you start to say, but your pixie is already gone. [[Return to the Harbor->"Well, see, you could just write down the number 6 and the fact that the 6 is in the hundred's place."]]

"Close," your pixie says. "How many zeros are there in 10,000,000?" "Seven," you say. "Well, that means that you need seven 10s multiplied together to make 10,000,000." [["Ah okay. Let me try again, then."->"Hmmm. . ."]]

"Close. Try one more time," your pixie says. "How many 8s are there in 8 x 8 x 8 x 8 x 8 x 8?" "It's hard to count," you say again more insistently. "They all blend together." "True," your pixie says. "Try counting it a few times, holding your finger or pencil over the nubers as you go. Try counting forwards and then backwards and make sure you get the same thing." "Okay . . . I think there are [[5]] [[6]] [[7->5]]

"Exactly! There are 6 eights. So, writing that using exponents would give us what?" [[Let me see . . .->"Hmm. I think I can try that . . ."]]

"Not quite," your pixie says again. "In 3^^4^^, the little floaty number, the superscript, tells us how many we're multiplying together. For example in 3^^9^^ there would be nine 3s multiplied together." "Like 3x3x3x3x3x3x3x3x3?" "Exactly. So, in 3^^4^^ how many 3s should there be?" [[Three]] [[Four]]

"Correct! You need to write four 3s all multiplied together. Like 3x3x..." [["Okay, let me try again."->"I can do that."]]

He looks skeptical but slightly hopeful. "How?" "You don't need all those zeros. Look!" [[Show him how to rewrite his numbers]]

(set: $answer to (prompt: "Rewrite 20,000 in scientific notation. Your answer should be in the form '2x10^#'", "")) (if: ($answer is "2x10^4"))[ (goto: "20,000 Correct") ] (else:)[ (goto: "20,000 Try Again") ]

Just one or two more people and you'll have it, you think. And there just happen to be a few more people around! One is a woman striding around her garden muttering to herself. The other is a man sitting inside his wagon, surrounded by wood shavings and scribbling with a short pencil on a sheet of paper. [[Approach the Woman in the Garden]] [[Approach the Man in the Wagon]]

The woman's hair is full of leaves and twigs, and a bee is sleeping on top of one ear. "Oh, hello," she says, stopping suddenly and smiling. [["Is something bothering you?" you ask.]]

"Oh wow!" he says. "That's incredible! That will save me so much time. I hate writing out super giant numbers." "You're welcome," you say, smiling. It's so great seeing math helping people in the real world. This is all fantastic, and you're sure you're close now. You almost have it. But you feel like there's one thing you're missing. Perhaps because it's so small . . . There's the woman pacing in her garden. You could talk to her if you haven't already, but you also notice a man you didn't see before. He's holding what looks like a very tiny telescope. [[Approach the Woman in the Garden]] [[Approach the Man with the Tiny Telescope]]

"Hmmm are you sure?" he asks. [["Oh, oops. Let me try again.->Show him how to rewrite his numbers]]

You approach the man with the tiny telescope. He appears to be using it to examine his own fingernail. His brow furrows in concern. [["Can I help you?" you ask.]]

She waves a hand, and several flower petals and a tuft of moss float out from somewhere in her sleeve. "Well, I've been doing some calculations about insects and trees. But, there are just so, so many of them! The numbers are just so enormous. They're hard to work with." Wow! You knew your discovery would help people, but you had never imagined it would help //so many// people! [["That's convenient!"]]

"Why do you say that?" she asks. "There's nothing convenient about 3,000,000,000,000 trees in the world with 200,000 leaves each. I've been trying to multiply them together to figure out the total number of leaves in the world. But, it's //so many zeros//! "You never have to write so many zeros again!" you shout. [[Show her how to write 3,000,000,000,000 in scientific notation.]]

(set: $answer to (prompt: "Rewrite 3,000,000,000,000 in scientific notation. Your answer should be in the form '3x10^#'", "")) (if: ($answer is "3x10^12"))[ (goto: "3trillion Correct") ] (else:)[ (goto: "3trillion Try Again") ]

"Oh wow! That is just wonderful!" she shouts. "That will make them so much easier to multiply, too!" You realize that she's right! Rather than multiplying 3,000,000,000,000 by 200,000, she could just multiply 3x10^^12^^ x 2x 10^^5^^ Multiplication can be done in any order, so that would be the same as 3 x 2 x 10^^12^^x10^^5^^ Multiply the first two numbers to get 6 And then you have 12 tens and 5 more tens, so 17 tens overall. So the answer is just 6x10^^17^^ So efficient! "Thank you so much!" she says, grinning. The bee smiles in its sleep. [["You're welcome!"]]

"Hmmm are you sure?" she asks. [["Oh, oops. Let me try again.->Show her how to write 3,000,000,000,000 in scientific notation.]]

This is all fantastic, and you're sure you're close now. You almost have it. But you feel like there's one thing you're missing. Perhaps because it's so small . . . There's the man sitting in his wagon, you could talk to him if you haven't already, but you also notice a man you didn't see before. He's holding what looks like a very tiny telescope. [[Approach the Man in the Wagon]] [[Approach the Man with the Tiny Telescope]]

"Oh. Well. I doubt it." "What are you working on?" you ask. "Well, I've been watching my fingernail grow for the last hour. I believe it has grown 0.00000417 meters." "Ah. Let me guess. It's an annoying number to work with?" His face breaks into a tentative smile. "Precisely! Are you telling me you have a solution?" All day you've been working with giant numbers. But you're sure that there's a way to apply your discovery to very tiny numbers, too. Wasn't there something about using exponents to divide instead of multilply. . .? [[Ask your pixie about using exponents for division]] [[Show the man how to write his number in scientific notation]]

"Pixie?!" you shout. Your pixie appears. Her hair is on fire and there's a strange green gooey substance dripping from one arm. "Yes?" "Er, are you okay?" "Yeah! Just make it quick, my cousin found a Flaming Slime Mold in his basement!" [["Well, can you explain how to divide using exponents?"]]

"It's no problem," you say. "You see, I've invented something called scientific notation! And it will let you write very tiny numbers very easily and efficiently, too!" "Really?" he asks, his eyes widening. [["Yes!"]]

(set: $answer to (prompt: "Rewrite 0.00000417 in scientific notation. Your answer should be in the form '4.17x10^#'", "")) (if: ($answer is "4.17x10^-6"))[ (goto: "fingernail Correct") ] (else:)[ (goto: "fingernail Try Again") ]

"Wow! That is just wonderful. Could I also use that for the width of my hair?" He holds out a tuft of bright red hair. "It's 0.000017 meters thick." [["Of course!"]]

"Oh. . . Are you sure?" the man asks. [["Oops, let me try again."->"Yes!"]]

(set: $answer to (prompt: "Rewrite 0.000017 in scientific notation. Your answer should be in the form '1.7x10^#'", "")) (if: ($answer is "1.7x10^-5"))[ (goto: "hair Correct") ] (else:)[ (goto: "hair Try Again") ]

"Absolutely incredible!" the man shouts, jumping up and clicking his heels together. He drops his tiny telescope but you dive and catch it. "Ah, thank you, thank you," he says, smiling at you, slightly out of breath. "You know. There's someone who needs to hear about this invention." "Who?" "My cousin Helen! She'll love this." "Helen? As in. . . Helen the Builder?" you ask, hardly daring to hope. "That's the one! Come on!" He grabs your hand. You can't believe your luck. All this time you've wanted to talk to her, and you thought you were getting father and farther away all the time. The dwarf and all the others you've helped come along, travelling with you all the way back to the great office of Helen the Builder. At last, just as evening falls, you arrive at her office. Your new friends knock on the door, and you are welcomed inside. You are led to a great hall with many long tables, and at the far end sits none other than the great Helen the Builder. "Helen," your new friend say. "You've got to hear this new math thing this person has invented." "Oh?" her eyebrows lift. "Well, I do have a lot to get done here. . . but. . . yes I'd love to hear. We can always make use of new math tools." [["Thank you so much! I'll be quick!"]]

"Oh. . . Are you sure?" the man asks. [["Oops, let me try again."->"Of course!"]]

"Of course! If you want to divide by the same number a bunch of times, just use a negative exponent instead of a positive one!" "So, if I were dividing by 2 seven times?" She glances over her shoulder "2^^-7^^." "I think so!" "So, in your case, if you wanted to divide by 100?" [["Oh, let me think . . ."]]

(set: $answer to (prompt: "Rewrite 0.01 using negative exponents. Your answer should be in the form '10^#'", "")) (if: ($answer is "10^-2"))[ (goto: "neg Correct") ] (else:)[ (goto: "neg Try Again") ]

"Yep!" your pixie says. You hear a faint shriek and an ominous glugging sound. She winces. [[Thanks for your help, I've got it from here!"->"Can I help you?" you ask. [["Can I try one more?"

"Not quite! Try again!" your pixie says. [[Okay hmmm->"Oh, let me think . . ."

"Great idea! What's 0.00001?" she asks. A jet of fire shoots over her shoulder from somewhere, but she dodges it. [["Written as an exponent? Let me see . . ."->"Okay!"]]

"Write it as a fraction first!" "Oh, you mean how 0.00001 is the same as 1/100,000?" "Yes!" [["Okay!"

(set: $answer to (prompt: "Rewrite 0.00001 using negative exponents. Your answer should be in the form '10^#'", "")) (if: ($answer is "10^-5"))[ (goto: "neg2 Correct") ] (else:)[ (goto: "neg2 Try Again") ]

"Perfect! Got to go!" She disappears. You hope her cousin is okay. . . [[Return to the man with the tiny telescope->Approach the Man with the Tiny Telescope]]

"Not quite! Try again!" your pixie says. [["Okay!"]]

</div> <button class="tiny-button"> (link: "💾")[(save-game: "MySaveSlot")(alert: "Save successful!")] (link: "📂")[(alert:"Loading your saved game now...")(load-game: "MySaveSlot")] </button>

(if: (saved-games:) contains "autosave")[ (link: "Continue")[(load-game: "autosave")] ]

"Of course! It looks weird, but it's not too bad." You wait a bit nervously. Sometimes your pixie forgets how complicated things can be. It's understandable though. Once we know how to do something, it seems easy to us and we forget how hard it was to learn the first time. Like shoe-tying, or talking. "Well," she says. "You know how, when we want to multiply a bunch of times, we use positive exponents?" "Yeah . . ." "Well, when we divide, we use negative exponents!" That doesn't seem too bad . . . "Like, if I wanted to divide by 3 twice, that would be 3^-2." "But that doesn't quite make sense," you say. "Because you can't just divide by 3. There has to actually be a number you're dividing, right? Like 18 divided by 3?" "Oh, right. Yes. Well, technically, we're dividing 1 by 3." [["Why the number 1? Why not some other number?"]]

"Great question! Very sharp-eyed!" your pixie says, smiling. "And doesn't that make the division type different than multiplication? Because 5^2 is just 5 x 5. Not 1 x 5 x 5. Although, I guess that would still be the same, wouldn't it?" "Yes, it would! If it helps, you can think of it as there always being a secret 1 there." "Is that allowed?" "Yes! And it actually fits with a cool pattern exponents make. Let me show you with tens." She holds up her wand and some sparkling numbers appear in the air. 10^^3^^ = 1,000 10^^2^^ = 100 10^^1^^ = 10 "Now, before I go on, I'll warn you," she says. "The first time someone showed me this I didn't really buy it. It didn't make sense. It's grown on me, though, and now I think it's cool. But it's okay if it doesn't make sense this first time." "That's a relief." "Yes. All you really need to know is that 10^^-1^^ means divide by 10 once, and 10^^-2^^ means divide by 10 twice, and so on. What would 10^^-4^^ be?" [["It would mean you'd divide by ten four times."]] [["I'm not sure"]]

"Exactly! That's where the 1 comes from. So, look, the rest of the pattern goes like this:" She waves her wand and even more number appear. You jump. It looks very complicated to you and you're glad that she said you don't really have to fully understand this part right off the bat. That all you need to know is that 10^^-5^^ for example just means 1 divided by 10 five times. 10^^3^^ = 1,000 10^^2^^ = 100 10^^1^^ = 10 10^^0^^ = 1 10^^-1^^ = 0.1 10^^-2^^ = 0.01 10^^-3^^ = 0.001 "See," she says breathlessly. "Just look at the numbers on the right. If you start at the top and go down, you just divide by 10 each time. 1,000 divided by 10 is 100. 100 divided by 10 is 10. 10 divided by 10 is 1. And then 1 divided by ten is 0.1." "Interesting," you say. "Wait, does that say 10^^0^^? How can you have zero tens multiplied together?" "That's a case where it makes more sense to think of it as 1 times zero 10s, doesn't it?" "Hmm . . . I guess I can see that." "Great!" your pixie says. You hear a faint shriek and an ominous glugging sound. She winces. [[Thanks for your help, I've got it from here!"->"Can I help you?" you ask.]] [["Can you give me a problem to try?"->"Can I try one more?"]]

"Precisely! You understand this well!" "Okay, well, then I'll just go back to—" "NO WAIT THIS IS SO COOL HANG ON ONE SECOND!" she shouts. Startled, you pause. She seems so excited that you suppose you'll listen. "See, here, imagine hopping from the bottow row up each row one-by-one to the top. To go from 10 to 100, you multiply by 10. To go from 100 to 1,000 you multiply by 10. Every time you go up a row, you multiply by 10." 10^^3^^ = 1,000 10^^2^^ = 100 10^^1^^ = 10 "Okay, yeah, I see that." "So then, to go down a row, you //divide// by ten." "Oh, huh. So . . . from 1,000 to 100 you divide by 10. From 100 to 10 you divide by 10." "Exactly. So, keep going with the pattern. If you divide 10 by 10, what do you get?" [["10 divided by 10 is 1!]]

"No worries at all! Thanks for letting me know!" Your pixies says. "10^^-4^^ would mean we'd divide by 10 four times." "Ah, okay, thanks." "What would 10^^-5^^ be?" [["Divide by 10 five times?"->"It would mean you'd divide by ten four times."]] [["Multiply by 10 negative five times?"]]

"A totally fair way to think about it!" your pixies says. "You can definitely think of dividing as undoing some multiplication you already did! Like 3 x 2 = 6. And 6 divided by 2 is 3." [["Weird!"->"It would mean you'd divide by ten four times."]]

The dwarf pauses and turns to look at you. [["1,000 could also be written as 10^3!"]] [["1,000 could also be written as 10^4!"]] [[Hmmm . . . these look weird, you decide to ask your pixie for help->Ask your pixie to remind you about how exponents work]]