Expand and Compress: Re-meeting Arithmetic

The Re-Frame

School arithmetic taught you that addition and multiplication are "operations." That's a thin description. A richer one: addition and multiplication expand; subtraction and division compress. Multiplication is denormalization. Division is normalization. Logs (you'll meet them in two tracks) are the deluxe normalizer.

This isn't a vocabulary game. It changes how you read code:

img / 255.0 isn't "division." It's compression — squeezing pixel values into [0, 1].
output * temperature isn't "multiplication." It's expansion — stretching the logit distribution.
x - mean isn't "subtraction." It's centering — shifting the origin to where the action is.

The Image Inspector Story

Dad's math-book has a tiny Streamlit app called Image Inspector. Loads an image. Divides pixel values by 255 (compress). Adjusts brightness by adding a slider value (shift). Clips to [0, 1] (compress again, into legal range). Multiplies by 255 to display (expand). All four operations are arithmetic. None of them are "math problems." They're shape adjustments.

Every arithmetic operation is a shape change. When you see x / k, ask: what was the shape, what's the new shape? The answer is more useful than the answer to "what's the value?"

Track Reward

The infinite is real. Sampling makes it tractable. Every operation reshapes; every shape costs something. You can now look at any AI preprocessing pipeline and read it as compress, expand, shift, clip, repeat. That's most of what data preprocessing actually is.

Code

Compress / shift / clip / expand — the whole pipeline·python

import numpy as np

img = np.random.randint(0, 256, (64, 64, 3)).astype(np.float32)

img_norm   = img / 255.0                            # compress to [0, 1]
img_bright = np.clip(img_norm + 0.2, 0.0, 1.0)      # shift, then clip
img_back   = (img_bright * 255).astype(np.uint8)    # expand to [0, 255]

print(img_back.dtype, img_back.min(), img_back.max())

Exercise

Write 3 lines of code that take an array of 100 ages (random integers 18-90), shift them so the mean is 0, scale them so they fit in [-1, 1]. Identify which line is *compression*, which is *centering*, and which is *clipping* (if any).

Hint

Two ops are enough — center then divide by max-abs. The 'clipping' is implicit if your max-abs catches everything.

Expand and Compress: Re-meeting Arithmetic

The Re-Frame

The Image Inspector Story

Track Reward

Code

Exercise

Progress

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