4.7 — Introduction to scientific notation

In this lesson, you'll learn about scientific notation, how it's used in mathematics and programming, and how C++ represents very large and very small numbers using this system.

What is scientific notation?

Scientific notation is a way to express very large or very small numbers in a compact, standardized form. Instead of writing many zeros, we use powers of 10 to represent the magnitude.

The general form is: a × 10^n

• a is called the mantissa or significand (typically between 1 and 10)
• n is the exponent (can be positive or negative)

Think of it like this: if regular numbers are like writing out the full address "123 Main Street, Apartment 4B, Springfield, State 12345", then scientific notation is like using a zip code - it conveys the same information more efficiently.

Examples of scientific notation

Large numbers

6,400,000,000 = 6.4 × 10^9
299,792,458 = 2.99792458 × 10^8  (speed of light in m/s)
602,214,076,000,000,000,000,000 = 6.02214076 × 10^23  (Avogadro's number)

Small numbers

0.000001 = 1 × 10^-6
0.0000000000164 = 1.64 × 10^-11
0.00000000000000000016 = 1.6 × 10^-19  (charge of an electron)

Scientific notation in C++

C++ uses a slightly different format for scientific notation in code:

• Instead of × we use e or E
• Instead of 10^n we write en or En

So 6.4 × 10^9 becomes 6.4e9 or 6.4E9 in C++.

#include <iostream>

int main()
{
    std::cout << "Scientific notation in C++:\n\n";
    
    // Large numbers
    double speedOfLight = 2.99792458e8;      // 299,792,458 m/s
    double avogadro = 6.02214076e23;         // Avogadro's number
    double earthMass = 5.972e24;             // Earth's mass in kg
    
    // Small numbers
    double planckConstant = 6.62607015e-34;  // Planck constant
    double electronCharge = 1.602176634e-19; // Elementary charge
    double protonMass = 1.67262192e-27;      // Proton mass in kg
    
    std::cout << "Large numbers:\n";
    std::cout << "Speed of light: " << speedOfLight << " m/s\n";
    std::cout << "Avogadro's number: " << avogadro << " particles/mol\n";
    std::cout << "Earth's mass: " << earthMass << " kg\n\n";
    
    std::cout << "Small numbers:\n";
    std::cout << "Planck constant: " << planckConstant << " J⋅Hz^-1\n";
    std::cout << "Electron charge: " << electronCharge << " C\n";
    std::cout << "Proton mass: " << protonMass << " kg\n";
    
    return 0;
}

Output:

Scientific notation in C++:

Large numbers:
Speed of light: 2.99792e+08 m/s
Avogadro's number: 6.02214e+23 particles/mol
Earth's mass: 5.972e+24 kg

Small numbers:
Planck constant: 6.62607e-34 J⋅Hz^-1
Electron charge: 1.60218e-19 C
Proton mass: 1.67262e-27 kg

Understanding the parts

Let's break down scientific notation step by step:

#include <iostream>

int main()
{
    std::cout << "Breaking down scientific notation:\n\n";
    
    // Example: 4.5e6 means 4.5 × 10^6
    double number = 4.5e6;
    
    std::cout << "4.5e6 breakdown:\n";
    std::cout << "Mantissa (significand): 4.5\n";
    std::cout << "Exponent: 6\n";
    std::cout << "Meaning: 4.5 × 10^6\n";
    std::cout << "Calculation: 4.5 × 1,000,000 = " << number << "\n\n";
    
    // Example: 1.23e-4 means 1.23 × 10^-4
    double smallNumber = 1.23e-4;
    
    std::cout << "1.23e-4 breakdown:\n";
    std::cout << "Mantissa (significand): 1.23\n";
    std::cout << "Exponent: -4\n";
    std::cout << "Meaning: 1.23 × 10^-4\n";
    std::cout << "Calculation: 1.23 × 0.0001 = " << smallNumber << "\n\n";
    
    // Show the relationship
    std::cout << "Understanding powers of 10:\n";
    std::cout << "10^3 = " << 1e3 << " (1,000)\n";
    std::cout << "10^2 = " << 1e2 << " (100)\n";
    std::cout << "10^1 = " << 1e1 << " (10)\n";
    std::cout << "10^0 = " << 1e0 << " (1)\n";
    std::cout << "10^-1 = " << 1e-1 << " (0.1)\n";
    std::cout << "10^-2 = " << 1e-2 << " (0.01)\n";
    std::cout << "10^-3 = " << 1e-3 << " (0.001)\n";
    
    return 0;
}

Output:

Breaking down scientific notation:

4.5e6 breakdown:
Mantissa (significand): 4.5
Exponent: 6
Meaning: 4.5 × 10^6
Calculation: 4.5 × 1,000,000 = 4.5e+06

1.23e-4 breakdown:
Mantissa (significand): 1.23
Exponent: -4
Meaning: 1.23 × 10^-4
Calculation: 1.23 × 0.0001 = 0.000123

Understanding powers of 10:
10^3 = 1000 (1,000)
10^2 = 100 (100)
10^1 = 10 (10)
10^0 = 1 (1)
10^-1 = 0.1 (0.1)
10^-2 = 0.01 (0.01)
10^-3 = 0.001 (0.001)

Converting between regular and scientific notation

Converting regular numbers to scientific notation

#include <iostream>
#include <iomanip>

void convertToScientific(double number)
{
    std::cout << std::fixed << "Regular: " << std::setprecision(0) << number;
    std::cout << std::scientific << " → Scientific: " << std::setprecision(2) << number << "\n";
}

int main()
{
    std::cout << "Converting to scientific notation:\n\n";
    
    std::cout << "Large numbers:\n";
    convertToScientific(123456789);
    convertToScientific(5000000);
    convertToScientific(98765.4321);
    
    std::cout << "\nSmall numbers:\n";
    convertToScientific(0.000123);
    convertToScientific(0.00000005);
    convertToScientific(0.000000000789);
    
    return 0;
}

Output:

Converting to scientific notation:

Large numbers:
Regular: 123456789 → Scientific: 1.23e+08
Regular: 5000000 → Scientific: 5.00e+06
Regular: 98765 → Scientific: 9.88e+04

Small numbers:
Regular: 0 → Scientific: 1.23e-04
Regular: 0 → Scientific: 5.00e-08
Regular: 0 → Scientific: 7.89e-10

Converting scientific notation to regular numbers

#include <iostream>

void showConversion(double scientificValue, const char* scientificStr)
{
    std::cout << "Scientific: " << scientificStr << " = " << scientificValue << "\n";
}

int main()
{
    std::cout << "Converting from scientific notation:\n\n";
    
    // Large numbers
    showConversion(1.5e6, "1.5e6");
    showConversion(2.0e3, "2.0e3");
    showConversion(7.25e9, "7.25e9");
    
    std::cout << "\nSmall numbers:\n";
    showConversion(3.2e-5, "3.2e-5");
    showConversion(1.0e-8, "1.0e-8");
    showConversion(4.567e-12, "4.567e-12");
    
    // Manual verification
    std::cout << "\nManual calculations:\n";
    std::cout << "1.5e6 = 1.5 × 1,000,000 = " << 1.5 * 1000000 << "\n";
    std::cout << "3.2e-5 = 3.2 × 0.00001 = " << 3.2 * 0.00001 << "\n";
    
    return 0;
}

Output:

Converting from scientific notation:

Scientific: 1.5e6 = 1.5e+06
Scientific: 2.0e3 = 2000
Scientific: 7.25e9 = 7.25e+09

Small numbers:
Scientific: 3.2e-5 = 3.2e-05
Scientific: 1.0e-8 = 1e-08
Scientific: 4.567e-12 = 4.567e-12

Manual calculations:
1.5e6 = 1.5 × 1,000,000 = 1.5e+06
3.2e-5 = 3.2 × 0.00001 = 3.2e-05

Practical applications

Physics and science calculations

#include <iostream>
#include <cmath>

int main()
{
    std::cout << "Scientific notation in physics calculations:\n\n";
    
    // Constants
    double c = 2.998e8;           // Speed of light (m/s)
    double h = 6.626e-34;         // Planck constant (J⋅s)
    double k_B = 1.381e-23;       // Boltzmann constant (J/K)
    
    // Calculate photon energy: E = h × f
    double frequency = 5.0e14;    // 500 THz (green light)
    double energy = h * frequency;
    
    std::cout << "Photon energy calculation:\n";
    std::cout << "Planck constant: " << h << " J⋅s\n";
    std::cout << "Frequency: " << frequency << " Hz\n";
    std::cout << "Energy: " << energy << " J\n\n";
    
    // Calculate kinetic energy at different temperatures
    std::cout << "Thermal energy at different temperatures:\n";
    double temperatures[] = {273.15, 310.15, 373.15};  // 0°C, 37°C, 100°C
    
    for (double T : temperatures)
    {
        double thermalEnergy = 1.5 * k_B * T;  // Average kinetic energy
        std::cout << "T = " << T << " K → E = " << thermalEnergy << " J\n";
    }
    
    return 0;
}

Financial and statistical calculations

#include <iostream>

int main()
{
    std::cout << "Large numbers in finance and statistics:\n\n";
    
    // GDP values (in USD)
    double usGDP = 2.1e13;        // ~$21 trillion
    double worldGDP = 8.5e13;     // ~$85 trillion
    
    // Population statistics
    double worldPopulation = 7.8e9;  // ~7.8 billion people
    double usPopulation = 3.3e8;     // ~330 million people
    
    std::cout << "Economic data:\n";
    std::cout << "US GDP: $" << usGDP << "\n";
    std::cout << "World GDP: $" << worldGDP << "\n\n";
    
    std::cout << "GDP per capita:\n";
    std::cout << "US GDP per capita: $" << usGDP / usPopulation << "\n";
    std::cout << "World GDP per capita: $" << worldGDP / worldPopulation << "\n\n";
    
    // Very small probabilities
    double lotteryOdds = 1.0 / 2.92e8;  // ~1 in 292 million
    std::cout << "Lottery winning probability: " << lotteryOdds << "\n";
    
    return 0;
}

Common mistakes and best practices

✅ Correct scientific notation

#include <iostream>

int main()
{
    // Correct ways to write scientific notation
    double correct1 = 1.5e6;      // Good
    double correct2 = 1.5E6;      // Also good (E or e both work)
    double correct3 = 1.5e+6;     // Explicit positive exponent (optional)
    double correct4 = 2.3e-4;     // Negative exponent
    
    std::cout << "Correct scientific notation:\n";
    std::cout << "1.5e6 = " << correct1 << "\n";
    std::cout << "1.5E6 = " << correct2 << "\n";
    std::cout << "1.5e+6 = " << correct3 << "\n";
    std::cout << "2.3e-4 = " << correct4 << "\n";
    
    return 0;
}

❌ Common mistakes

// These are syntax errors:
// double wrong1 = 1.5 × 10^6;   // Can't use × or ^
// double wrong2 = 1.5 * 10^6;   // ^ is not exponentiation in C++
// double wrong3 = 1.5x10^6;     // x is not valid

// These are mathematically correct but inefficient:
double inefficient1 = 1.5 * 1000000;        // Works but verbose
double inefficient2 = 1.5 * pow(10, 6);     // Works but unnecessary

✅ When to use scientific notation

#include <iostream>

int main()
{
    // Use scientific notation for very large numbers
    double avogadro = 6.022e23;          // Much clearer than 602200000000000000000000
    
    // Use scientific notation for very small numbers
    double electronMass = 9.109e-31;     // Much clearer than 0.0000000000000000000000000000009109
    
    // Don't use scientific notation for moderate numbers
    double price = 19.99;                // Not 1.999e1
    double temperature = 25.5;           // Not 2.55e1
    
    std::cout << "Good use of scientific notation:\n";
    std::cout << "Avogadro: " << avogadro << "\n";
    std::cout << "Electron mass: " << electronMass << " kg\n\n";
    
    std::cout << "Regular notation for moderate values:\n";
    std::cout << "Price: $" << price << "\n";
    std::cout << "Temperature: " << temperature << "°C\n";
    
    return 0;
}

Summary

Scientific notation is essential for representing extreme values efficiently:

Format: a × 10^n in mathematics becomes aen or aEn in C++

• Mantissa/Significand: The decimal number (usually 1-10)
• Exponent: The power of 10 (positive for large numbers, negative for small)

When to use:

• Very large numbers (> 1,000,000)
• Very small numbers (< 0.001)
• Scientific calculations
• When exact representation of extreme values is needed

Benefits:

• Compact representation
• Easier to read and compare magnitudes
• Reduces errors from counting zeros
• Standard in scientific computing

Scientific notation helps you work with the full range of values that computers can represent, from subatomic particles to astronomical distances.

Quiz

1. How do you write 1,250,000 in C++ scientific notation?
2. What does 3.7e-5 equal in decimal form?
3. What are the two parts of scientific notation called?
4. When should you use scientific notation instead of regular decimal notation?
5. Is 1.5E6 the same as 1.5e6 in C++?

Practice exercises

Try these scientific notation exercises:

1. Convert these regular numbers to scientific notation: 45,000,000 and 0.000078
2. Convert these scientific notation values to regular form: 2.5e4 and 6.1e-7
3. Write a program that calculates the number of atoms in a mole using Avogadro's number
4. Create a physics calculation program that uses scientific notation for constants like the speed of light and Planck's constant