Why is 0! = 1 and a0 = 1 for a = 0? The case must be excluded a = 0?

Mathematics, science and Society in connection with mathematical writing

The accounting must be mathematical and simple and accurate. Think through how you use headings, different ways to write formulas and so on. Any pictures should be made in some program and not drawn by hand.  It  is of course not the right to copy any text from the other or any book, but you should also go through and formulate your own answers.

1. Why is 0! = 1 and a0 = 1 for a = 0? The case must be excluded a = 0?
2. What is 2/3 ? Why is 2/3 = 4/6 ? How would you respond if the person who asks is a student in grade 6? How would you answer if it was a question on an exam in basic mathematics? In school, fractions are usually introduced as part of something and that is probably how many perceive them, but they are actually  numbers from a mathematical point of view. How would you explain what p/q is, regarded as a number?
3. Explain why  a/b * c/d = ac/bd,   but do not lie  from the abstract construction of the rational numbers with equivalence classes and so on.
4. Discuss what is good and less good about the following x2 − 3x + 2 = 0:

If  x2 −3x+2 = 0 ⇒ (x−3/2)2 −(3/2)2 +2 = 0, so  (x− 3/2)2 − 1/4 = 0 ⇔ x − 3/2 = ±1/2. Roots are also x = 3/2 ± 1/2 ⇒ x = 1 ∨ x = 2.

• Förklara skillnaden mellan ett ordnat par och en mängd med 2 element.
• Förklara skillnaden mellan en ekvation och en identitet.
• Är det någon skillnad mellan en funktion och dess graf? Vari består den i så fall?

Note: Task 5, 6 and 7 are inspired by Vivaldi’s book.