🔢 NumPy Mathematical Functions: Your Magic Calculator Toolbox

Imagine you have a super-powered calculator that can do math on thousands of numbers at once. That’s NumPy! Today, we’ll explore its magical math tools.

🌟 The Big Picture

Think of NumPy as a kitchen full of cooking tools. Just like you have:

• A measuring cup (constants like π)
• A timer (exponential growth/decay)
• A rounding spoon (rounding numbers)
• A spinning wheel (trigonometry for circles)
• A mirror box (complex numbers with real and imaginary parts)

NumPy gives you all these math tools, but for millions of numbers at once!

📐 NumPy Constants: Numbers That Never Change

Some numbers are so special, they have names. NumPy keeps them ready for you!

Meet the Famous Constants

import numpy as np

# Pi - the circle number
print(np.pi)        # 3.14159...

# e - the growth number
print(np.e)         # 2.71828...

# Infinity - bigger than anything
print(np.inf)       # inf

# Not a Number - for impossible math
print(np.nan)       # nan

🎯 Why Do We Need Them?

Simple Example:

# Circle with radius 5
radius = 5
area = np.pi * radius ** 2
print(f"Area: {area}")
# Output: Area: 78.539...

📈 Exponential and Log Functions: Growth and Shrink Magic

The Story of Exponential Growth

Imagine you have 1 bunny. Each day, every bunny has 2 babies.

• Day 0: 1 bunny
• Day 1: 2 bunnies
• Day 2: 4 bunnies
• Day 3: 8 bunnies

This is exponential growth – things multiply super fast!

Exponential Functions

import numpy as np

# e raised to power (natural exponential)
print(np.exp(1))      # 2.718... (e^1)
print(np.exp(2))      # 7.389... (e^2)

# 2 raised to power
print(np.exp2(3))     # 8 (2^3 = 2×2×2)

# Any base raised to power
print(np.power(2, 4)) # 16 (2^4)

Logarithm: The Reverse Button

If exponential is multiplication on steroids, logarithm is the undo button.

# Natural log (base e)
print(np.log(np.e))     # 1.0

# Log base 2
print(np.log2(8))       # 3.0 (2^3 = 8)

# Log base 10
print(np.log10(1000))   # 3.0 (10^3 = 1000)

🎯 Quick Reference

graph TD
    A[Number] --> B{Operation?}
    B --> C[exp: Grow fast]
    B --> D[log: Find the power]
    C --> E[Small → Big]
    D --> F[Big → Power used]

🔄 Rounding and Absolute Value: Making Numbers Neat

Rounding: Tidying Up Messy Numbers

Think of rounding like cleaning your room – you put things in the nearest right place.

import numpy as np

messy = np.array([1.2, 2.5, 3.7, -1.4])

# Round to nearest integer
print(np.round(messy))
# [1. 2. 4. -1.]

# Always round DOWN (floor)
print(np.floor(messy))
# [1. 2. 3. -2.]

# Always round UP (ceil)
print(np.ceil(messy))
# [2. 3. 4. -1.]

# Chop off decimals (truncate)
print(np.trunc(messy))
# [1. 2. 3. -1.]

🎯 Rounding Cheat Sheet

Absolute Value: Distance from Zero

Absolute value is like asking “How far from zero?” – direction doesn’t matter!

numbers = np.array([-5, -2, 0, 3, 7])

# Get absolute values
print(np.abs(numbers))
# [5 2 0 3 7]

# Also works with fabs for floats
print(np.fabs([-3.5, 4.2]))
# [3.5 4.2]

Real Life: Temperature difference doesn’t care about direction. Whether it’s 5° warmer or 5° colder, the change is 5°!

🎡 Trigonometric Functions: The Circle Math

The Ferris Wheel Story

Imagine sitting on a Ferris wheel. As it spins:

• Your height goes up and down smoothly → that’s sine!
• Your distance from the center goes back and forth → that’s cosine!

import numpy as np

# Angles in radians
angle = np.pi / 4  # 45 degrees

# Sine - height on circle
print(np.sin(angle))  # 0.707...

# Cosine - horizontal distance
print(np.cos(angle))  # 0.707...

# Tangent - slope of line
print(np.tan(angle))  # 1.0

Converting Degrees ↔ Radians

# Degrees to radians
deg = 90
rad = np.radians(deg)
# or np.deg2rad(deg)
print(rad)  # 1.5707... (π/2)

# Radians to degrees
rad = np.pi
deg = np.degrees(rad)
# or np.rad2deg(rad)
print(deg)  # 180.0

Inverse Trig: Finding the Angle

# If sin(angle) = 0.5, what's angle?
print(np.arcsin(0.5))    # 0.523... rad
print(np.degrees(
    np.arcsin(0.5)))     # 30.0 degrees!

# Same for others
print(np.arccos(0.5))    # 1.047... rad
print(np.arctan(1.0))    # 0.785... rad

🎯 Trig Function Summary

graph LR
    A[Angle θ] --> B[sin θ: Height]
    A --> C[cos θ: Width]
    A --> D[tan θ: Slope]
    E[Value] --> F[arcsin: Find angle]
    E --> G[arccos: Find angle]
    E --> H[arctan: Find angle]

🔮 Complex Number Operations: Numbers with Imagination

What Are Complex Numbers?

Some equations have no “real” answer. Like: What number times itself equals -1?

We invented i (imaginary unit) where: i² = -1

A complex number has two parts:

• Real part: Normal number
• Imaginary part: Number × i

import numpy as np

# Create complex number: 3 + 4i
z = 3 + 4j  # Python uses 'j' not 'i'
# or
z = np.complex64(3 + 4j)

print(z)  # (3+4j)

Working with Complex Numbers

z = 3 + 4j

# Get the real part
print(np.real(z))      # 3.0

# Get the imaginary part
print(np.imag(z))      # 4.0

# Absolute value (magnitude)
print(np.abs(z))       # 5.0 (like 3-4-5 triangle!)

# Angle (phase) in radians
print(np.angle(z))     # 0.927... rad

# Complex conjugate (flip sign of i)
print(np.conj(z))      # (3-4j)

🎯 Complex Number Visual

Think of complex numbers as arrows on a treasure map:

graph LR
    A[Origin 0,0] --> B[3 + 4j]
    subgraph Parts
        C[Real: 3 steps right]
        D[Imaginary: 4 steps up]
        E[Magnitude: 5 total distance]
    end

🎁 Putting It All Together

Here’s a mini program using everything we learned:

import numpy as np

# Constants
print(f"Circle area (r=2): {np.pi * 2**2}")

# Exponential & Log
print(f"e^3 = {np.exp(3):.2f}")
print(f"log2(16) = {np.log2(16)}")

# Rounding
print(f"Round 3.7 = {np.round(3.7)}")
print(f"Abs of -5 = {np.abs(-5)}")

# Trigonometry
angle = np.radians(45)
print(f"sin(45°) = {np.sin(angle):.3f}")

# Complex
z = 3 + 4j
print(f"|3+4j| = {np.abs(z)}")

Output:

Circle area (r=2): 12.566...
e^3 = 20.09
log2(16) = 4.0
Round 3.7 = 4.0
Abs of -5 = 5
sin(45°) = 0.707
|3+4j| = 5.0

🌈 Remember This!

You did it! 🎉 You now have a full toolbox of NumPy math functions. These aren’t just for homework – they power video games, weather prediction, music apps, and rockets to space!