Numbers Beyond the Classroom

Consider the number 5. When you square it, you get 25. If you square 25, you get 625. Notice the pattern? The square of 5 ends in 5, the square of 25 ends in 25, and the square of 625 ends in 625. Does this pattern continue? Let's try squaring 390,625. While it doesn't end exactly in 390,625, the last 5 digits match, extending the pattern. If we square just that part (90,625), it does end in itself. By repeating this process, we can keep finding numbers that are their own squares. These numbers have infinitely many digits and are equal to their own squares.

You might question the notion of numbers with infinite digits to the left of the decimal point, approaching infinity. In this video, I aim to convince you that such numbers do make sense. They belong to a number system that operates differently from our familiar one. These numbers are vital in cutting-edge research fields like number theory and algebraic geometry.

Let's explore the properties of the number system that includes these numbers. We'll call them "10-adic numbers" because they are written in base 10. Can we add two 10-adic numbers together? Absolutely! You add them digit by digit from right to left, just like regular addition. What about multiplication? Again, you can multiply any two 10-adic numbers. The last digit of the answer depends only on the last digits of the numbers being multiplied, and the subsequent digits depend on the numbers to their right. Although calculations may become extensive, you can continue indefinitely to as many digits as needed.

For instance, consider a 10-adic number ending in 857,142,857,143 and multiply it by 7. The answer can be obtained digit by digit. This shows that this number multiplied by 7 equals 1, indicating that this 10-adic number represents 1/7. Interestingly, we can find other rational numbers within the realm of 10-adic numbers without using the division symbol.

Let's find the 10-adic number equivalent to 1/3. We can imagine an infinite string of digits where multiplying them by 3 yields 1. This implies that all the digits to the left of 1 must be 0. To determine the digits multiplying by 3, we start with the unit's place. 3 times 7 is 21, giving us the 1 and carrying the 2. Then, we find what to multiply by 3 plus 2 to get a 0. It turns out to be 6. Therefore, an infinite string of 6s with one 7 to the left represents 1/3. This resembles the familiar infinite decimal digits like 0.999 repeating, which is equal to 1 in the conventional decimal system.

Similarly, if we consider a 10-adic number composed of infinite 9s to the left of the decimal point, it can be set as variable m. By multiplying both sides by 10, we find that this infinite string of 9s equals -1. Adding 1 to it results in every digit becoming 0. Although a 1 might seem inevitable on the left side, the infinite string of 9s prevents it. Therefore, all 9s combined equal -1. This discovery reveals that the 10-adic numbers include negative numbers without requiring additional symbols.

In summary, 10-adic numbers can be added, subtracted, and multiplied just like we expect. They even accommodate fractions and negative numbers without the need for extra symbols. However, there is a challenge when we examine the first 10-adic number we foundwhich is its own square. Rearranging the terms, we end up with n times (n - 1) equals 0. While 0 and 1 satisfy this equation, our 10-adic number is neither 0 nor 1. This poses a challenge as mathematicians often rely on rearranging terms, setting them equal to 0, and factoring equations to solve them. This approach breaks down with 10-adic numbers because we are working in base 10, which is a composite number (5 multiplied by 2).

Suppose we want to find two 10-adic numbers that multiply to 0. We know that the last digit must be 0. We can multiply 0 by any number, but we can also multiply 5 by 4 to get 20, resulting in a 0 in the unit's place. By carrying the 2 and continuing this process, we can construct numbers where all the digits become 0. However, there is a solution to this issue: using a base of a prime number instead of a composite one.

In conclusion, numbers beyond what we learn in school can be fascinating and intriguing. The realm of 10-adic numbers introduces us to a different number system where infinite digits to the left of the decimal point have meaning. These numbers possess unique properties and offer new perspectives in various areas of mathematical research. While they may challenge our traditional methods of solving equations, they open up doors to exploring mathematics in a different light.