Python and Math Foundations with Runnable Examples

What you need

These exercises use Python 3 and its standard library: no account, paid API, or third-party package is needed. Save each complete code block to its own file and run it with python3 filename.py.

Learn functions, lists, dictionaries, files, exceptions, and basic tests first. SQL helps you select training examples; HTTP and JSON help you integrate models into an application. A web developer can keep a TypeScript frontend while using Python for model experiments.

Features are measurable inputs

A delivery might have distance, number of items, and rain as features. A vector is an ordered list of numbers. The order matters: swapping rain and distance changes the meaning.

features = [8.0, 3.0, 1.0]  # distance_km, item_count, rain_indicator
weights = [2.0, 0.5, 4.0]
bias = 5.0

prediction = sum(x * w for x, w in zip(features, weights)) + bias
print(prediction)  # 26.5 minutes

This is a linear model with illustrative weights chosen by us. Training would estimate weights from examples. The dot product multiplies matching components and adds them; the bias supplies an offset.

Normalize using training data

Features measured in very different units can make some optimization methods harder to use. Standardization transforms a value using a training mean and standard deviation.

from statistics import mean, pstdev

training_distances = [2.0, 4.0, 6.0, 8.0]
center = mean(training_distances)
scale = pstdev(training_distances)

def transform(distance):
    return (distance - center) / scale if scale else 0.0

print(round(transform(6.0), 3))  # 0.447
print(round(transform(10.0), 3))  # 2.236

Keep center and scale with the model. Use those same values for future data. Recomputing them using test data or each incoming batch changes the representation and can introduce leakage.

Compare vectors with cosine similarity

Cosine similarity compares direction. It divides a dot product by both vector lengths. For nonzero vectors, its range is from -1 to 1. It is a geometric score, not a probability of correctness.

from math import sqrt

def cosine(left, right):
    if len(left) != len(right):
        raise ValueError("Vector dimensions must match")
    dot = sum(a * b for a, b in zip(left, right))
    lengths = sqrt(sum(a * a for a in left)) * sqrt(sum(b * b for b in right))
    return dot / lengths if lengths else 0.0

print(round(cosine([1, 1, 0], [1, 0, 0]), 3))  # 0.707
print(cosine([1, 0], [0, 1]))  # 0.0

Real text embeddings are learned representations. These hand-written vectors demonstrate the calculation without pretending to provide semantic understanding.

Count classification mistakes

Suppose a fraud detector raises five alerts, of which three are actual fraud, and misses one additional fraudulent payment.

true_positive = 3
false_positive = 2
false_negative = 1

precision = true_positive / (true_positive + false_positive)
recall = true_positive / (true_positive + false_negative)
print(precision, recall)  # 0.6 0.75

Precision answers how many alerts are correct. Recall answers how much actual fraud was found. In production, handle zero denominators explicitly and report results for relevant classes and groups.

What math should you learn next?

Try it yourself

Change the weights in the delivery example and explain why the prediction changes. Add a zero-vector case to the cosine example. Build a confusion matrix for a small labeled dataset and explain why high accuracy could hide poor recall.

Key takeaways

• Features have names, units, and an order that must remain consistent.
• Fit preprocessing on training examples and reuse it at inference.
• Similarity, probability, and evaluation scores mean different things.