# Sara wanted a bed time story

Sara was really attached to her grandmother Shanti. She had called her Nana ever since she could say her first words. Nana would often tell a story or sing a lullaby before Sara would sleep. Even when Sara became a teenager, occasionally she would insist that Nana tell her a bedtime story. This was one of those days.

Sara: Nana, tell me a story please, a long one please.

Nana: A story about what? Sara, I have told you all the stories I know.

Sara: Anything. Tell me a story about you when you were a little girl living in India.

Nana: A long story about me when I was a little girl – let me think. Okay, I I will tell you a story. It’s kind of funny though. My mom told me most of it.

Sara: I love funny stories. Please, tell me already.

# Story when Nana was little

Nana: Our family was neither very rich nor very poor when I was born. We had enough to live on. We also had a relative named Shah. He was a cousin of my mother Sheela. I called him uncle when I started to talk. He lived in the USA. Whenever, he came to visit us he would boast about him being very rich. My mother told me that uncle Shah was at our home when I was born.

Shah: Sheela, I am happy that you gave me a niece. I want to give her a big gift.

Sheela: Your being here with us on this occasion is a big gift for us.

Shah: Sheela please, let me give her a gift. How does $10,000 sound?

Sheela: I don’t want you to give $10,000, boast about it now and then forget us. No, it’s not right.

Shah: How about if I give her $1000 every year for the next 10 years?

Sheela was very smart and told uncle: I’ll tell you what, just give her one dollar.

Shah: One dollar! That’s an insult to Shanti.

Sheela: Okay then. Give her one dollar now, and then double it the next year.

Shah: Does that mean two dollars each year then onwards? That’s too little.

# Give her 1 dollar now, and double it every year

Sheela: God give you all the wealth. If this is too little, just keep doubling it each year as long as you and Shanti live.

Shah: Even after she is married?

Sheela: You don’t want to break the relationship with her after she is married, do you?

Shah: I am sorry. I will give her double the amount each year for as long as Shanti and I are alive.

Sheela: Don’t forget your promise. May God give you more and more so that you can keep your generous promise.

Uncle Shah didn’t think that the gift was very generous considering that his wealth was around one billion dollars. I don’t know whether my dad was happy or angry that my mother didn’t take the $10,000 as a gift from uncle Shah. He never told me that.

I am told that uncle Shah gave me a dollar and promised to come the next year. My mom got a piggy bank for me and put this dollar in it.

I was one year old when he came the next time. I had just started to talk. I played with him and called him uncle. He was very happy. He gave me two dollars before he left. Now my piggy bank had three dollars – one dollar from before plus the two he gave me on my first birthday.

The next year, he didn’t come but he sent me four dollars which was double the previous time. I was only two years old.

Uncle Shah came to our place again when I was three. By now, I began to understand that he cared for me. I danced around him all day and chatted with him. Again, he gave me 8 dollars which was double of the amount he had given me last time.

Then he didn’t come for my next three birthdays but he kept his promise. He sent me 16, 32 and 64 dollars for each of the three years. I didn’t know much but my mom thought that was a lot of money for a 6 year old. She told me not to tell my friends because they would be jealous, and may start to hate me.

I heard what my mom said and just went out to play stapu with one of my friends. Stapu is what you call hopscotch.

You know, I had learned to add and subtract by this time. I came home and started thinking. Uncle Shah gave me $64. By now, I could calculate what he would give me the next year.

He gave me $128, $256 and $512 on my next three birthdays. Here I was a 9 year old and my piggy bank was way more than full. My mom didn’t put the money in a bank because our local bank didn’t deal in foreign currencies. She decided to put the money in my pillow case. My pillow case had all this money and my pillow. I used to sleep with all the money every night. Can you believe that (giggle)?

# She kept $8191 in her pillow case

Uncle Shah kept his promise and gave me $1024, $2048 and $4096 in the next three years. By now I was 12 years of age – a preteen as you might say. I counted it all and I had $8191 in my pillow case.

My mom decided that this was too much money to be kept at home. She asked my dad to make some arrangement. My dad and I went to a big city nearby where my dad opened an account in US dollars. He deposited the money there. The bank wouldn’t give us any interest on foreign currencies. He still thought it was better this way because a large amount of money at home could attract robbers and that would be risky for our lives.

Remember, at my next birthday I would be thirteen. I am told that I was beginning to blossom. I proudly sent my picture on the birthday to uncle Shah. He sent me the promised money which was $8192. My dad deposited it in the bank along with the money from before. Only my dad handled all the money. I was not even old enough to have a joint account with him.

I overheard my mom and dad talking about uncle Shah. It seems his wife, aunt Rose, was getting suspicious because uncle Shah was giving so much money regularly to a young girl. I even overheard that his marriage could be in peril because of it. I told my mom that I felt guilty about it.

# Shah regularly gave a lot of money to a young girl

Uncle Shah and aunt Rose came to our house on my fourteenth birthday. There was a big birthday party. Many of my friends and some of our relatives were invited. After the others left, uncle Shah handed me the next doubled amount which was a hefty sum of $16384. I refused to take it because I was feeling guilty already. I told him to keep it. He got very upset and started talking to my mom.

Shah: Sheela, what is this? I thought we had an agreement.

Sheela: I don’t want to keep this agreement any more. You have given her enough already.

Shah: Why so?

Sheela: I think your wife feels awkward that you are giving a large sum of money to my beautiful young girl.

Shah: I see where this is going. Okay, accept it this last time and no more from next year onwards.

Aunt Rose: Sheela. Don’t break his heart. Tell Shanti to keep the money. We can easily afford it. Also, I don’t feel awkward about it.

Sheela: Okay Rose. This would be the last time, and that also on one condition. Not Shah but you hand the money to Shanti so that it doesn’t look awkward to her. Aunt Rose then came over to me, gave me a big hug and the money.

I got married at the age of 16 and kept that money in a different bank account where they paid interest. I still have all of it. I don’t spend it because your mom and dad give me enough for all my expenses.

Sara told Nana that she must have been a very beautiful girl for aunt Rose to get jealous. Memories of those teenage days brought tears in Nana’s eyes. Sara went to sleep but kept dreaming about the bedtime story from Nana.

# Sara told the story to her boyfriend

Next morning, she walked to school with her boyfriend Johnny. She told him the story. It would be an understatement to say that Johnny was overwhelmed. After school, they went to Johnny’s home for a short time. They loved being together.

Johnny: Is uncle Shah still alive?

Sara: No, he died twenty years ago.

Johnny: So that means Nana was 40 years old then. I wonder how much he would have paid her in his last year if aunt Rose didn’t get mad at him, and if he had continued giving the money doubling each year. You said that he gave Nana $16384 when she turned 14. That means, he doubled the one dollar amount 14 times.

# Increasing exponentials

Sara: My mom explained to me the best way to understand that is as an exponential function f(t) = a^{t} which has “a” as the base and the power t, In this case the base is 2 which is the ratio by which it increases with the power t which is the number of years. Thus f(14) = 2^{14}. So he gave her $2^{14} when she was 14 and would have given her f(40) = 2^{40} dollars when Nana was 40 years old. According to my calculations that would have been $1,099,511,627,776.

Johnny: That’s a lot of money. It is over a thousand billion dollars. None of the richest men have that much money. Any of the rich men has only less than one hundred billion. That means if aunt Rose didn’t get mad, uncle Shah would have broken his promise. Wow! So uncle Shah keeping his promise was just a fantasy. As long as we are doing this, can we fantasise that he remained alive and would have kept paying Nana even now when she is 60. How much would he have to pay for this year?

# Rules for exponentials

Sara: Our algebra book has this law for exponents. It says that a^{n }x a^{m} = a^{(n+m)}.

Because 60 = 20 + 40, we could write m = 40 and n = 20 and m + n = 60.

Then 2^{60} = 2^{40} +2^{20}.

We know that 2^{40} = $1,099,511,627,776. My calculator says that 2^{20} = 1,048,576. We can just multiply these two numbers using my computer.

Then $2^{60} = $1,099,511,627,776 x 1,048,576. That would be over one million trillion dollars.

Johnny: I don’t think that my calculator can handle that kind of multiplication and give me an exact answer. I am impressed.

Sara: Here is something that would impress you even more. We could have done all this without using a calculator. You know that 2 x 2 x 2 x 2 x 2 = 32. I multiplied 32 by 32 on a piece of paper and it came out to 1024. Thus, 2^{10} = 1024 which is approximately 1000.

Jonny: I see where you are going. You are going to use the rule a^{n }x a^{m} = a^{(n+m)}. Using the same rule for the exponents six times, you will write that because 60 =10 + 10 +10 +10 +10 +10,

2^{60} = 2^{10} x 2^{10} x 2^{10} x 2^{10} x 2^{10} x 2^{10} or about 1000 x1000 x1000 x1000 x1000 x1000 =1,000,000,000,000,000,000 which is approximately $1,099,511,627,776 x 1,048,576. That’s good. I never thought of doing it that way.

# What if he quadrupled the money every year?

Johnny paused for a bit and then said: All this stuff is fantasy, anyway. I want to say that uncle Shah was kind enough to say that he would give quadruple the money for Nana every year instead of doubling it. How many years would it take for him to reach the same amount as he gave by doubling in the year 14 ?

Sara: Here is another rule in our book we can use. (a^{n})^{m} = a^{nm}. We know that 4 = 2^{2}.

Johnny: That’s nice. We can say (2^{2})^{m} = 2^{2m} or 4^{m} = 2^{2m} or 4^{7} = 2^{2 x 7}. That means, uncle Shah would have given Nana the same amount by the quadrupling rule after 7 years as he did after 14 years by the doubling rule.

Johnny was glad to figure this out but not quite satisfied, and said: This was too easy because we know that 2^{2} = 4. So let me ask how many years would it take for him to reach the same amount if he was to give triple the amount each year. So how do we use this exponent rule there?

Sara: I guess we use the same rule . (a^{n})^{m} = a^{nm }but we have to solve for 2^{n} = 3 first. We can go over how to solve for n in this case later but let’s see if the rule works. I will check if someone on the Internet gives us this value.

Here it is 2^{1.584963} = 3.

Then 2^{14} = (2^{1.584963})^{m }= 2^{1.584963m} or 1.584963m = 14 or m =14/1.584963 = 8.833. That means it would have taken 8.833 years but because he gave her the money only on her birthdays, it would mean he would have given her less than 2^{14 }on the 8th birthday and more than that on the 9th. Hey, that seems right because this number is between 7 and 14 years.

Johnny: All that was fun but what do exponents have to do with reality?

Sara: Johnny. I think it’s time to call it the day. Why don’t we look through our algebra book? I will also talk to my mom and dad. We can talk about it another time.

With this Sara went home but they met the next day again.

# Applications of exponentials

Sara: Johnny, I talked to my parents and looked in our algebra book There are lots of applications of exponential functions in real life. They are used for population studies including bacterial population in lakes, and in radioactive decay studies. However, the most important application is in financial Math. You know how they talk about simple and compound interest rates. Like, if you deposit the money in a bank for two years and it gives you a 10% interest rate. Writing an amount of 1 for the principal and 0.1 for the interest per year, for simple interest rate the amount will increase to 1 + 0.1 x 2 or 1.2 but with compound interest rate it will become 1.1^{2} which is 1.21. The differences keep getting bigger with the increasing number of years. Like, my Nana put $1000 in a long term deposit at 10% per annum interest rate and I can cash it when I turn 18. With the simple interest rate this would have grown to 1000 +1000 x 18 x 0.1 which is $2800 but with the compound interest rate it will be 1000 x 1.1^{18} which about $5600. That’s a big difference.

Johnny: Yes, that’s a big difference. I guess, that’s why mortgages cost a lot.

Sara: Economists also use exponential functions to study inflation. So, I guess Nana told me a good bedtime story.

*Challenge*

Tom and Melissa have been dating secretly and didn’t want anyone to know. By mistake Melissa just told Chatty Maddie at school about it at 9 am. She didn’t realize that Maddie was a rumour machine. Within 1 hour Maddie had told three of her friends. These friends also liked to share the rumour and each of them told three of their friends within the next hour. The rumour machine continued running the same way. How many people, do you think knew about it by 3 pm when Melissa left the school to go home?

* Solution: *At 9 am, when Melissa told Maddy, only 3 people knew about it – Tom, Melissa and Maddy. Let us call it the time zero. At 10 am which is at the time hour 1, 3 more people knew, let’s call that 3 x 3^{1. } Now, at 11 am which is hour 2, the number of people who knew about it increased to 3 x 3^{2}. At this rate by 3 pm which would be hour 6, the number of people who expected to know about it was 3 x 3^{6 }which is 3 x 3 x 3 x 3 x 3 x 3 x 3. Now because 3 x 3 x 3 = 27, you could quickly calculate that 3 x 27 x 27 people knew about it. This comes out to 2187. Well you know how many students are in your school. Do you think the rumour also spread outside your school? Can you imagine this going on for one full day which is 24 hours? Figure it out. I think that the even Chatty Maddy’s rumour machine would have to slow down because there are not that many people in this world.