Arithmetic & Number Systems for AI

Integers

1. Definition

An integer is a whole number without a decimal point. It can be positive, negative, or zero.

2. Principle

Integers represent countable quantities.

If something can be counted as a whole item, it is usually stored as an integer.

Quantity	Integer Example
Epochs	10
Batch Size	32
Number of GPUs	4
Number of Features	128
Token Count	512

Rule: Counts = Integers

3. Properties of Integers

Property	Example
Positive	8
Negative	-5
Zero	0
No decimal	25 ✓
Decimal is not integer	25.5 ✗

4. Visual Number Line

The diagram above is the number line for integers:

• Left side → Negative integers
• Middle → Zero
• Right side → Positive integers
• The dots … mean the line continues forever in both directions

5. Mathematical Operations

6. Python Implementation

# Integers
epochs = 10
batch_size = 32
gpu_count = 4
max_tokens = 512

print(type(epochs))      # <class 'int'>
print(type(batch_size))  # <class 'int'>

Arithmetic

epochs = 10
extra_epochs = 5

total = epochs + extra_epochs
print(total)  # 15

7. AI & Machine Learning Use Cases

Example 1 — Training Epochs

epochs = 20

The model will learn from the dataset 20 times.

Example 2 — Batch Size

batch_size = 64

The neural network processes 64 images at once.

Example 3 — Vocabulary Size

vocab_size = 50000

The tokenizer contains 50,000 words/tokens.

Example 4 — Token Limit

max_tokens = 2048

An LLM can generate up to 2048 tokens.

8. Real-World AI Examples

AI System	Integer Used
ChatGPT	Max Tokens = 4096
CNN	Batch Size = 32
BERT	Sequence Length = 512
YOLO	Number of Classes = 80
Training	Epochs = 100

9. Common Mistakes

Wrong

batch_size = 32.5

Batch size cannot be half a sample.

Correct

batch_size = 32

Wrong

epochs = 10.0

Correct

epochs = 10

10. Interview Questions

Q1. What is an integer?

A whole number without a decimal point.

Q2. Why is batch size an integer?

Because it represents the number of training samples processed together, and samples are countable.

Q3. Is zero an integer?

Yes. Zero belongs to the integer set.

Q4. Is 7.0 an integer?

Mathematically it represents 7, but in programming 7.0 is a float, not an integer.

11. Practice Questions

1. Is -12 an integer?
2. Is 0 an integer?
3. Is 18.5 an integer?
4. Which is correct for epochs: 20 or 20.0?
5. Why can't batch size be 16.5?

Answers: 1. Yes · 2. Yes · 3. No · 4. 20 · 5. Samples cannot be fractional.

12. One-Line Revision

Integers are whole numbers used to represent countable values such as epochs, batch size, GPUs, features, and token counts in AI systems.

Formula Sheet

Memory Rule: Count = Integer

Floats

1. Definition

A float (floating-point number) is a number that contains a decimal point. It is used to represent values that are not whole numbers.

Examples: 0.5, 3.14, 0.001, -2.75

2. Principle

Floats represent measurements, probabilities, and continuous values.

If a value can vary smoothly (instead of being counted), it is usually a float.

Quantity	Float Example
Learning Rate	0.001
Dropout	0.25
Temperature	0.8
Confidence Score	0.97
Loss Value	0.142

Rule: Measurements = Floats

3. Properties of Floats

Property	Example
Decimal number	3.14
Positive	0.75
Negative	-1.25
Zero can be float	0.0
Can represent fractions	0.5

Integer vs Float

Integer	Float
5	5.0
32	32.5
-7	-7.25
0	0.0

Although 5 and 5.0 have the same value, they are different data types.

4. Visual Number Line

5. Mathematical Operations

6. Python Implementation

# Float variables
learning_rate = 0.001
dropout = 0.25
temperature = 0.8
loss = 0.142

print(type(learning_rate))   # <class 'float'>

Arithmetic

loss1 = 0.45
loss2 = 0.32

total_loss = loss1 + loss2
average_loss = total_loss / 2

print(average_loss)

7. AI & Machine Learning Use Cases

Example 1 — Learning Rate

learning_rate = 0.001

Controls how much the model updates its weights during training.

Example 2 — Dropout

dropout = 0.3

30% of neurons are randomly disabled during training to reduce overfitting.

Example 3 — Temperature

temperature = 0.7

Controls randomness in LLM responses.

• Low = More deterministic
• High = More creative

Example 4 — Confidence Score

confidence = 0.98

Model is 98% confident in its prediction.

8. Real-World AI Examples

AI System	Float Used
Adam Optimizer	0.001
GPT Temperature	0.7
CNN Dropout	0.5
BERT Learning Rate	2e-5
Model Confidence	0.96

9. Common Mistakes

Wrong

learning_rate = 1

A learning rate of 1 is usually too large and causes unstable training.

Correct

learning_rate = 0.001

Wrong

temperature = 7

Correct

temperature = 0.7

10. Interview Questions

Q1. What is a float?

A decimal number used to represent continuous values.

Q2. Why is learning rate always a float?

Because weight updates require very small decimal values for stable optimization.

Q3. Difference between 5 and 5.0?

5	5.0
Integer	Float
Count	Measurement
int	float

Q4. Give three AI parameters that are floats.

• Learning Rate
• Dropout
• Temperature

11. Practice Questions

1. Is 0.25 a float?
2. Is 7.0 a float?
3. Which data type should learning rate use?
4. Is 32.5 suitable for batch size?
5. Why is 0.001 better than 1 for learning rate?

Answers

1. Yes
2. Yes
3. Float
4. No
5. It provides small, stable weight updates.

12. One-Line Revision

Floats are decimal numbers used to represent continuous values such as learning rates, probabilities, loss values, dropout, and temperature in AI systems.

Formula Sheet

Parameter	Typical Value
Learning Rate	0.001
Dropout	0.25
Temperature	0.7
Confidence	0.98

Memory Rule: Measure = Float

Fractions

1. Definition

A fraction represents a part of a whole. It consists of two integers separated by a horizontal line.

Examples:

• Numerator → Top number (parts selected)
• Denominator → Bottom number (total equal parts)

2. Principle

Fractions represent proportions or portions of a complete set.

In AI, fractions are commonly used for dataset splitting, probabilities, and sampling.

Fraction	Meaning
1/2	Half
1/4	One quarter
3/4	Three quarters
4/5	Four parts out of five

Rule: Part of a whole = Fraction

3. Parts of a Fraction

Example:

• Numerator = 3
• Denominator = 4

Meaning: 3 of the 4 equal parts are selected.

4. Visual Representation

1/2

3/4

5. Converting Fractions

Fraction → Decimal

Fraction → Percentage

Table

Fraction	Decimal	Percentage
1/2	0.50	50%
1/4	0.25	25%
3/4	0.75	75%
4/5	0.80	80%

6. Mathematical Operations

7. Python Implementation

from fractions import Fraction

train_split = Fraction(4, 5)
print(train_split)

# Output: 4/5

Decimal conversion

train_split = 4 / 5
print(train_split)

# Output: 0.8

8. AI & Machine Learning Use Cases

Example 1 — Train/Test Split

Suppose a dataset has 1000 images.

• Training = 800
• Testing = 200

total_images = 1000

train = int(total_images * (4/5))
test = total_images - train

print(train, test)

Output:

800 200

Example 2 — Validation Split

Dataset = 10,000 samples

Fraction	Samples
7/10	7000
2/10	2000
1/10	1000

This is a 70/20/10 dataset split.

Example 3 — Probability

A model predicts:

• Cat = 3/4
• Dog = 1/4

Converted to percentages:

• Cat = 75%
• Dog = 25%

9. Real-World AI Examples

AI Task	Fraction
Dataset Split	4/5
Validation	1/10
Sampling	1/100
Mini Dataset	1/20
Train Ratio	7/10

10. Common Mistakes

Wrong

Writing dataset split as:

train = 4
test = 5

This is not a fraction.

Correct

train_ratio = 4 / 5

Wrong

Using fractions that don't total 1.

• Train = 3/4
• Test = 1/2

Total = 1.25 ❌

Correct

• Train = 4/5
• Test = 1/5

Total = 1 ✅

11. Interview Questions

Q1. What is a fraction?

A number representing a part of a whole.

Q2. What are numerator and denominator?

• Numerator → Selected parts
• Denominator → Total parts

Q3. Why are fractions used in ML?

To divide datasets into training, validation, and testing portions.

Q4. Convert 3/4 into decimal and percentage.

• Decimal = 0.75
• Percentage = 75%

12. Practice Questions

1. Convert 1/2 to decimal.
2. Convert 3/5 to percentage.
3. A dataset has 500 samples. Find the training size using 4/5.
4. Which is larger: 2/3 or 3/5?
5. Why must train + test fractions equal 1?

Answers

1. 0.5
2. 60%
3. 400
4. 2/3
5. Because they must represent the entire dataset.

13. One-Line Revision

Fractions represent parts of a whole and are used in AI for dataset splitting, probabilities, sampling, and proportional calculations.

Formula Sheet

Memory Rule: Part of Whole = Fraction

Percentages

1. Definition

A percentage means "per hundred". It expresses a value as a fraction of 100.

Examples: 10%, 50%, 80%, 95%, 99.8%

2. Principle

Percentages measure performance, proportions, and probabilities.

In AI, percentages are used to evaluate how well a model performs.

AI Metric	Example
Accuracy	95%
Precision	92%
Recall	88%
Confidence	97%

Rule: Performance = Percentage

3. Percentage Formula

Example

A model correctly predicts 95 images out of 100.

4. Visual Representation

80%

5. Convert Between Formats

Percentage	Fraction	Decimal
25%	1/4	0.25
50%	1/2	0.50
75%	3/4	0.75
80%	4/5	0.80
95%	95/100	0.95

Conversion Formula

Percentage → Decimal

Example:

Decimal → Percentage

Example:

6. Python Implementation

Calculate Percentage

correct = 95
total = 100

accuracy = (correct / total) * 100
print(accuracy)

Output:

95.0

Convert Decimal to Percentage

confidence = 0.97

print(confidence * 100)

Output:

97.0

7. AI & Machine Learning Use Cases

Example 1 — Accuracy

Suppose a classifier predicts:

• Correct = 980
• Total = 1000

correct = 980
total = 1000

accuracy = (correct / total) * 100
print(f"Accuracy: {accuracy}%")

Example 2 — Confidence Score

confidence = 0.93

Displayed to users as:

93%

Example 3 — Dataset Split

1000 images

• Train = 80%
• Test = 20%

Split	Samples
Train	800
Test	200

8. Real-World AI Examples

AI Application	Percentage
Image Classifier Accuracy	96%
Spam Detection Precision	91%
Face Recognition Confidence	99%
OCR Accuracy	98.5%
LLM Evaluation Score	87%

9. Common Mistakes

Wrong

accuracy = 95 / 100
print(accuracy)

Output:

0.95

This is a decimal, not a percentage.

Correct

accuracy = (95 / 100) * 100
print(accuracy)

Output:

95.0

Wrong

Treating 95 as a decimal.

confidence = 95

Correct

confidence = 0.95

10. Interview Questions

Q1. What does percentage mean?

A value expressed per hundred.

Q2. Write the percentage formula.

Q3. Difference between 0.95 and 95%?

Value	Meaning
0.95	Decimal
95%	Percentage

Both represent the same quantity.

Q4. Why are percentages important in ML?

They communicate model performance in an easily understandable way.

11. Practice Questions

1. Convert 60% to decimal.
2. Convert 0.85 to percentage.
3. A model predicts 180 correct out of 200. Find accuracy.
4. Which is larger: 92% or 0.87?
5. Why is confidence usually shown as a percentage?

Answers

1. 0.60
2. 85%
3. 90%
4. 92%
5. It is easier for humans to interpret.

12. One-Line Revision

Percentages express values per hundred and are used in AI to measure accuracy, precision, recall, confidence, and overall model performance.

Formula Sheet

Memory Rule: Performance = Percentage

Ratios

1. Definition

A ratio compares two or more quantities and shows their relationship.

Examples: 1:1, 2:1, 4:5, 16:9, 80:20

Key idea: A ratio compares values; it does not represent a part of a whole by itself.

2. Principle

Ratios express comparison between quantities.

In AI, ratios are used for dataset splits, image dimensions, class balance, and model evaluation.

Ratio	Meaning
80:20	Train : Test
70:15:15	Train : Validation : Test
16:9	Image aspect ratio
3:1	Majority vs minority class

Rule: Comparison = Ratio

3. Ratio Formula

If two quantities are A and B:

Example:

• Apples = 8
• Oranges = 2

This is called simplifying a ratio.

4. Visual Representation

Train : Test = 80 : 20

5. Simplifying Ratios

Example

Divide both sides by 4:

Another example:

Original	Simplified
12:4	3:1
80:20	4:1
100:50	2:1

6. Ratio vs Fraction vs Percentage

Type	Example	Purpose
Ratio	80:20	Compare
Fraction	4/5	Part of whole
Percentage	80%	Per hundred

Conversion

Ratio → Fraction

Ratio → Percentage

7. Python Implementation

Dataset Ratio

train = 80
test = 20

ratio = train / test

print(ratio)

Output:

4.0

Convert Ratio to Samples

total = 1000

train = int(total * (80/100))
test = total - train

print(train, test)

Output:

800 200

8. AI & Machine Learning Use Cases

Example 1 — Dataset Split

Dataset = 5000 images

Ratio = 80:20

Split	Samples
Train	4000
Test	1000

Example 2 — Train / Validation / Test

Ratio = 70:15:15

Dataset = 10,000

Split	Samples
Train	7000
Validation	1500
Test	1500

Example 3 — Image Aspect Ratio

A monitor has:

• Width = 1920
• Height = 1080

This is the standard widescreen ratio.

Example 4 — Imbalanced Dataset

Suppose:

• Cats = 900
• Dogs = 100

This tells us the dataset is imbalanced.

9. Real-World AI Examples

AI Task	Ratio
Train/Test	80:20
Train/Val/Test	70:15:15
Image Resolution	16:9
Class Imbalance	9:1
Sampling	1:10

10. Common Mistakes

Wrong

Confusing ratio with percentage.

80:20 = 80%

Incorrect.

Correct

• Ratio = 80:20
• Fraction = 4/5
• Percentage = 80%

Wrong

Using ratios that don't match the total.

Example:

Train = 70

Validation = 20

Test = 20

Total = 110 ❌

Correct

11. Interview Questions

Q1. What is a ratio?

A comparison between two or more quantities.

Q2. Difference between ratio and fraction?

Ratio	Fraction
Compares values	Represents part of whole
80:20	4/5

Q3. Why is 80:20 commonly used in ML?

It provides a good balance between training data and evaluation data.

Q4. What does 16:9 represent?

An image or screen aspect ratio.

12. Practice Questions

1. Simplify 40:20.
2. Convert 80:20 into a fraction.
3. Convert 3:1 into a percentage of the first value.
4. A dataset has 600 samples using 4:1 ratio. Find train and test samples.
5. Which represents comparison: Ratio or Fraction?

Answers

1. 2:1
2. 4/5
3. 75%
4. Train = 480, Test = 120
5. Ratio

13. One-Line Revision

A ratio compares two or more quantities and is used in AI for dataset splits, aspect ratios, class balance, and proportional comparisons.

Formula Sheet

Memory Rule: Compare = Ratio

Scientific Notation

1. Definition

Scientific notation is a compact way to write very large or very small numbers using powers of 10.

Where:

• a = coefficient (1 ≤ a < 10)
• 10 = base
• n = exponent (positive or negative)

Examples

Scientific	Decimal
1 × 10³	1,000
5 × 10²	500
1 × 10⁻³	0.001
1 × 10⁻⁵	0.00001

Rule: Very small or very large numbers = Scientific Notation

2. Principle

Instead of writing long decimals like:

AI engineers write:

or in Python:

1e-8

This improves readability, precision, and consistency.

3. Formula

General Formula

Positive Exponent

Move the decimal right.

Negative Exponent

Move the decimal left.

4. Visual Representation

Positive Exponents

Negative Exponents

Memory Rule

• Positive exponent → Move decimal Right
• Negative exponent → Move decimal Left

5. Conversion Table

Scientific	Decimal
1e1	10
1e2	100
1e3	1,000
1e6	1,000,000
1e-1	0.1
1e-2	0.01
1e-3	0.001
1e-5	0.00001
1e-8	0.00000001

Note: In Python, 1e-5 means 1 × 10⁻⁵.

6. Python Implementation

Writing Scientific Notation

learning_rate = 1e-3
epsilon = 1e-8
weight_decay = 5e-4

print(learning_rate)
print(epsilon)

Output:

0.001
1e-08

Verify Values

print(1e-5)
print(2.5e3)

Output:

0.00001
2500.0

7. AI & Machine Learning Use Cases

Example 1 — Learning Rate

learning_rate = 1e-3

Equivalent to:

This controls how much weights are updated during training.

Example 2 — Adam Optimizer

epsilon = 1e-8

Used to prevent division by zero in optimization algorithms.

Example 3 — Weight Decay

weight_decay = 5e-4

Equivalent to:

Used for regularization to reduce overfitting.

Example 4 — BERT Fine-Tuning

Typical learning rates:

Scientific	Decimal
5e-5	0.00005
3e-5	0.00003
2e-5	0.00002

These values appear in almost every Transformer training script.

8. Real-World AI Examples

Parameter	Value
CNN Learning Rate	1e-3
BERT LR	2e-5
GPT Fine-Tuning LR	5e-5
Adam Epsilon	1e-8
Weight Decay	5e-4

9. Common Mistakes

Wrong

learning_rate = 0.0000000001

Hard to read.

Correct

learning_rate = 1e-10

Wrong

Confusing 1e-3 with 10^-3.

1e-3 means:

Not:

10. Interview Questions

Q1. What is scientific notation?

A compact way to represent very large or very small numbers using powers of 10.

Q2. What does 1e-5 mean?

Q3. Why do AI engineers prefer 1e-5?

Because it is easier to read, less error-prone, and commonly used in ML frameworks.

Q4. Where is scientific notation used in AI?

• Learning rates
• Adam optimizer
• Weight decay
• Numerical stability constants

11. Practice Questions

1. Convert 1e3 to decimal.
2. Convert 5e-4 to decimal.
3. Write 0.00001 in scientific notation.
4. Which is larger: 1e-3 or 1e-5?
5. Why is 1e-8 used in Adam?

Answers

1. 1000
2. 0.0005
3. 1e-5
4. 1e-3
5. To improve numerical stability and avoid division by zero.

12. One-Line Revision

Scientific notation represents extremely large and small numbers using powers of 10 and is widely used in AI for learning rates, optimizers, and numerical stability.

Formula Sheet

Memory Rule: Positive exponent → Right · Negative exponent → Left

Logarithms

1. Definition

A logarithm answers one question:

"To what power must the base be raised to produce a number?"

Where:

• b = Base
• x = Number
• y = Exponent

Example

Because:

Logarithm	Answer
log₂(8)	3
log₁₀(100)	2
log₂(16)	4
ln(e)	1

2. Principle

Logarithms measure exponential growth, information, and uncertainty.

In AI they are fundamental to:

• Cross-Entropy Loss
• Information Theory
• Decision Trees
• Language Models
• Entropy

Rule: Information = Logarithm

3. Relationship Between Exponents & Logarithms

They are inverse operations.

Exponential	Logarithm
2³ = 8	log₂(8) = 3
10² = 100	log₁₀(100) = 2

4. Types of Logarithms

Base 2

Used in:

• Entropy
• Information theory
• Decision trees

Base 10

Used in:

• Scientific calculations
• Orders of magnitude

Natural Log

Used in:

• Machine learning optimization
• Cross-entropy
• Probability distributions

5. Logarithm Graph

As x increases, the logarithm grows slowly.

Example:

x	log₂(x)
2	1
4	2
8	3
16	4
32	5

6. Python Implementation

Natural Log

import math

print(math.log(10))

Base 2

import math

print(math.log2(8))

Output:

3.0

Base 10

import math

print(math.log10(1000))

Output:

3.0

7. AI & Machine Learning Use Cases

Example 1 — Cross-Entropy Loss

The most important loss function in classification.

Where:

• y = True label
• ŷ = Predicted probability

Suppose:

• True = 1
• Prediction = 0.9

Smaller loss = Better prediction.

Example 2 — Decision Tree Entropy

Entropy measures disorder.

Example:

• Cat = 50%
• Dog = 50%

Higher entropy = More uncertainty.

Example 3 — Language Models

LLMs predict token probabilities.

Token	Probability
cat	0.70
dog	0.20
bird	0.10

The loss is calculated using log probabilities.

8. Real-World AI Examples

AI System	Logarithm Used
GPT	Cross-Entropy
BERT	Log Loss
Decision Trees	Entropy
CNN Classifier	Negative Log Likelihood
Reinforcement Learning	Log Probability

9. Common Mistakes

Wrong

Thinking:

Correct

Wrong

Ignoring the base.

Incorrect unless the base is 2.

Correct

10. Interview Questions

Q1. What is a logarithm?

A logarithm finds the exponent required to produce a number.

Q2. Why are logarithms important in ML?

Because they measure information and are used in loss functions like Cross-Entropy.

Q3. Difference between log₂ and ln?

log₂	ln
Base 2	Base e
Entropy	Optimization

Q4. Where is logarithm used in GPT?

During Cross-Entropy Loss for next-token prediction.

11. Practice Questions

1. Find log₂(16).
2. Find log₁₀(1000).
3. Which log is used in entropy?
4. Which loss function uses logarithms?
5. Why does a correct prediction produce lower log loss?

Answers

1. 4
2. 3
3. Base 2
4. Cross-Entropy
5. Because high predicted probability gives a smaller negative log value.

12. One-Line Revision

Logarithms measure exponents, information, and uncertainty, making them essential for Cross-Entropy Loss, entropy, and language model training.

Formula Sheet

Memory Rule: Information = Logarithm