Imagine writing two versions of the same computer program. The first represents its integers as 32-bit binary numbers. The second writes the numbers in base 10, ASCII strings with each byte used to store one digit.
The second version has its upsides. Thirty-two bit numbers max out at several billion, but you can keep tacking digits onto the string until you’re out of memory.
That said, the program that uses 32-bit integers runs faster because it uses the native architecture of the CPU. The CPU was designed with this more compact format for numbers in mind, with special-purpose circuits like 32 bit adders.
The same principle applies to using one’s brain: Some things the brain can do quickly and intuitively, and some things the brain has to emulate using many more of the brain’s native operations. Sometimes thinking in metaphors is a good idea, if you’re human.
In particular, visualizing things is part of the brain’s native architecture, but abstract symbolic manipulation has to be learned. Thus, visualizing mathematics is usually a good idea.
When was the last time you made a sign error?
When was the last time you visualized something upside-down by mistake?
I thought so.
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