Cantor normal form multiplication

In summary, A possible cantor normal form for an ordinal a is a sum of the form a = {\omega^{\beta_1}}c_1 + {\omega^{\beta_2}}c_2 + ... and it can be shown that a multiplied by omega is equal to {\omega^{\beta_1+1}}. The proof involves the fact that the c_{i}'s are positive integers and the descending order of the \beta_{i} terms in the sum.
  • #1
daniel_i_l
Gold Member
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Let's say that a is an ordinal and it's cantor normal form is:
[tex] a = {\omega^{\beta_1}}c_1 + {\omega^{\beta_2}}c_2 + ... [/tex]
I read that
[tex] a \omega = {\omega^{\beta_1+1}} [/tex]
But I couldn't find a proof anywhere.
Can someone give me a source or point me in the right direction so that I can prove it myself?
Thanks.
 
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  • #2
The [tex]c_{i}'s[/tex] are positive integers, so [tex]c_{i}\omega=\omega[/tex] and:

[tex]\left(\omega^{\beta_{i}}c_{i}\right)\omega=\omega^{\beta_{i}}\omega

= \omega^{\beta_{i+1}}[/tex]

But the [tex]\beta_{i}[/tex] are in descending order, so their sum is equal to the largest element, which is [tex]\omega^{\beta_{1}+1}[/tex]
 

Related to Cantor normal form multiplication

1. What is Cantor normal form multiplication?

Cantor normal form multiplication is a mathematical operation used to multiply two numbers written in their Cantor normal form, which is a unique way of representing natural numbers using only powers of two. It involves multiplying the exponents of the two numbers and adding them together to get the exponent of the product, while the base remains the same.

2. How is Cantor normal form multiplication different from traditional multiplication?

Cantor normal form multiplication is different from traditional multiplication because it follows a different set of rules. In traditional multiplication, we multiply each digit of one number with each digit of the other number and add the results together. In Cantor normal form multiplication, we multiply the exponents and add them together, while the base remains the same.

3. What is the purpose of using Cantor normal form multiplication?

The purpose of using Cantor normal form multiplication is to simplify the calculation of large numbers and to make it more efficient. This method is especially useful for multiplying large numbers that are written in their Cantor normal form.

4. Can Cantor normal form multiplication be used for other types of numbers?

Yes, Cantor normal form multiplication can be used for any numbers written in their Cantor normal form, including negative numbers and fractions. It can also be extended to other types of numbers, such as complex numbers, by using a similar method.

5. Are there any limitations to using Cantor normal form multiplication?

One limitation of using Cantor normal form multiplication is that it can only be used for numbers written in their Cantor normal form. It is not suitable for multiplying numbers written in their standard form. Additionally, it may become more complicated for larger numbers with multiple digits in their exponents.

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