# Strange Math 2.4

So...
My teacher found out that I was receiving online help with my math, and let's just say she isn't very happy about it. During today's class, we were learning about converting between different bases, but when I got my assignment, she wouldn't tell me which base to convert into. She said that I should know how to do it with my online help. Can you figure out what base I am converting to, and what the answer to the last question should be?

$$9186_{10}=2300$$
$$741_{10}=200000$$
$$79075_{10}=134000$$
$$31200_{10}=79$$
$$7696_{10}= \ ?$$

Hint:

As a note, solving these problems is really a science.

Hint 2:

This, as you probably know, is not true base conversion. However, Excel may sometimes make the mistake my teacher did.

• To confirm, does $9186_{10} = 2300$ mean that the number $n$ represented as $9186$ in base $10$ is also represented as $2300$ in the unknown base $b$? Nov 1, 2021 at 19:25
• Correct. If it seems ambiguous, could you clarify how I should fix it? : ) Nov 2, 2021 at 15:38
• I think it’s fine. :) But I’m definitely stumped! Nov 2, 2021 at 16:15
• I minorly updated the hint to try an be more helpful. Good Luck! : ) Nov 2, 2021 at 16:21

$$7696_{10} = 10000000000$$

Reasoning

First convert the numbers on the left from decimal to hexadecimal.

$$9186 \rightarrow 23e2$$
$$741 \rightarrow 2e5$$
$$79075 \rightarrow 134e3$$
$$31200 \rightarrow 79e0$$

Notice now how the numbers on the right could also be interpreted (for example, by Excel) as decimal numbers written in scientific notation, i.e,

$$23e2 \rightarrow 23 \times 10^2 = 2300$$
$$2e5 \rightarrow 2 \times 10^5 = 200000$$
$$134e3 \rightarrow 134 \times 10^3 = 134000$$
$$79e0 \rightarrow 79 \times 10^0 = 79$$

In particular,
$$7696 \rightarrow 1e10 \rightarrow 1 \times 10^{10} = 10000000000$$

• Congratulations! Wonderfully explained and perfectly correct, as well. : ) Nov 5, 2021 at 11:41