Start Learning Numbers - Part 8 - Integers (Multiplication)

Thanks to all supporters! They are mentioned in the credits of the video :)

This is my video series about Start Learning Numbers. I hope that it will help everyone who wants to learn about the construction of numbers. This is something one needs as a good foundation for all advanced mathematics.

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I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[madaaz]

It's great to know that the relationship between integer numbers and natural numbers is not obvious.

[aza-joru]

Very good work and excellent explication. This definition of relative numbers is known as "jensen construction" and it is similar in spirit to the construction of vectors in the space.

[gnikola]

Question. In previous episodes we got to the natural numbers and 0, basically {0, 1, 2, ...}. Now, we got to the integers, {..., -2_z, -1_z, 0_z, 1_z, 2_z, ...}. We also know that N_0 is a proper subset of Z. However, don't we have to prove the equalities n_z = n, say, 0_z = 0, 1_z = 1 and so on? Based on the "box construction" I don't immediately see this identities, although it's clear they share wanted properties.

[nicolenatsai]

I have been trying to follow how you show something is well defined. What is it exactly

[homology]

Oh, this can be proved by "construction" from you. but the construction is suggested by an aximatic defion. So, I learn that the prove of (-1)(-1)=1 as a ring member. Like,
(-1)(-1)=(-1)(-1)+(-1)+1
=(-1)(-1)+(-1)1+1
=(-1)(-1+1)+1
=(-1)0+1
=(-1)0+(-1)0+(-(-1)0)+1
=(-1)(0+0)+(-(-1)0)+1
=(-1)0+(-(-1)0)+1
=1
is this correct proof?