It's a story of a child after a long time.

It's a story of a child after a long time.
I talked about this a little while ago over dinner with my child for about 30 minutes, and I personally want to record it, so I'm organizing it.

Explain the background first.

My child is in the sixth grade. He doesn't give good deeds or hard education, but just goes to an English math academy in the neighborhood in front of elementary school.
However, the child quickly followed the academy class, gradually outpacing the grade, and now he is learning middle school 2 math for about 2 years. Classes are not difficult for children, but I think I've heard a lot from people around me that after the exponential law part you're learning now, you have to learn a system of equations, and that's difficult. I've been talking about my worries for a while, and I've been telling you not to worry because I'll help you.

This evening, he brought up the equation again. And he said again that it's okay, but a system of equations is difficult, so I showed him a thumbnail of (-5)^x=5 YouTube that I looked up a while ago.

"You can solve these equations with an exponent."

The child looked surprised.

"It's beyond high school. It's not difficult, but it doesn't matter. I'll teach you later if you need it."

The child thought for a moment and asked.

"I only learned that the exponent is a natural number, so the x in here is not a natural number?"

"No, Jisoo doesn't have to be a natural number. Didn't you learn that?"

"I didn't learn it."

The child seemed confused. The child asked again.

"Then, could your index be negative?"

I said it.

"You know that the index is a natural number, right?"

"I know that."

"You know that multiplying numbers adds up to the index, right?"

"Of course I know that"

"That's it for Jisoo. I'll show you an example. Did you learn the case where the Jisoo is zero?"

"No. Could Jisoo be zero?"

"Let's think it can be done. What would 2^0 be, for example?"

"I don't know. 0?"

"Let's put it this way. The square of 2 is 2^2 = 2^(2+0) right?"

"Yes."

"Then the exponent is multiplied by the total number, so 2^(2+0) = 2^2 x 2^0, right?"

"I see"

"Then what is 2^0?"

The child thought for a moment and answered.

“1?”

"That's right. The power of 2 is 1"

"Wow, that's amazing."

"Then what's the zero power of 3? Think the same thing"

The child said right away.

"That's also 1?"

"Right. If you think about it the same way, the zeros of all numbers will all be one, right?"

"Oh, I see."

"Then let's consider the case where the exponent is negative. What is the zero power of 2?"

"It's 1"

"You can write 0 as 2-2?"

"Yes"

"Then it will be 2^0 = 1 = 2^(2-2) = 2^2 x 2^(-2) = 4 x 2^(-2) right?"

It was a little too much for my child to follow, so I sang the ceremony step by step and remembered it in my head.

"That's right."

"Then what is 2^(-2)?"

“1/4?"

"That's right. The reciprocal of the square of two. So if the exponent is negative, it's the reciprocal of the exponent being squared by a positive number of the same magnitude. So what is 2^(-3)?"

The child answered right away.

“1/8“

"Right. Nothing much, right?"

The kid was amazed, so I decided to talk a little bit more.

"What will happen to your fractional index? Can you guess?"

"No, I have no idea"

"Now, let's think of two again. Two is the power of two, right?"

"Yes"

"1" is 1/2 plus 1/2 right

"Yes"

"The exponent is multiplied by the total number, so 2 = 2^(1/2) x 2^(1/2) right? Then if 2^(1/2) is squared, it becomes 2, so what is 2^(1/2)?"

"Root 2?"

"That's right. You can think of fractional indices as root numbers. Then you've extended the exponent to rational numbers. You got natural numbers, zero, negative numbers, and fractions just by exponential law, right? I don't know the translation because this is a word I don't learn in high school, but it's called Analytical continuation in math. You don't have to remember words, and you can think of it as expanding the scope with these simple rules. Much of the math you're going to learn in the future is about extending what you've learned to Analytical continuity. It's not difficult to remember that the same simple rules apply consistently. It wasn't hard to explain now, was it?"

"Yes."

"Actually, all the mathematics you learn up to high school can be done with just a few simple principles. I'm afraid the equation will be difficult now, but you learned linear equations, right? All the solutions you'll learn in the future, whether they're simultaneous or quadratic, are ways to convert them all into linear equations. So you're changing it to x-a = 0. You know that x in this equation is a, right?"

"Of course I know that."

"Whether it's a joint equation or a quadratic equation, it's very difficult and complicated, so you might not know why you're doing this as you learn, but the goal is one. Transforming it into a linear equation like x-a = zero. Everything you learn from now on is about that, and don't be scared because I'll explain it to you if you need it."

"So the formula of the root of the quadratic equation is also to change it to x-a = 0?"

"Right. You know it's hard and complicated, right? But it's just a way to turn it into x-a = 0."

"Oh, I see."

Then the child asked.

"But dad, I told you earlier. What's the mistake?"

Think for a second about how to explain it. I just decided to go with the normal way.

"What he's explaining now is that you don't learn until high school. But it's not difficult, so listen carefully. Mistakes are straight."

"Straight?"

"Yes. Think of it as the same as a straight line. It's called a one-on-one response, and the nature of the real number is a straight line. Now let's think of cutting this straight line with a knife. There must be a number that corresponds to where the knife is cut off, right?"

"Yes"

"If you could express that number in fractions, what would it be?"

"A number that can be expressed in fractions."