Exponents and scientific notation

Why this matters

Technical work spans enormous ranges. A data center stores petabytes (about 10^15 bytes), a transistor gate is a few nanometres wide (around 10^-9 m), and one mole of a substance contains about 6.02 × 10^23 particles. Writing those numbers out in full is slow and error-prone. Exponents and scientific notation let you write, compare and multiply them in seconds.

Exponents

An exponent says how many times to multiply a base by itself:

• 2^3 = 2 × 2 × 2 = 8
• 10^4 = 10 × 10 × 10 × 10 = 10,000
• 5^1 = 5, and any non-zero number to the power 0 is 1 (10^0 = 1)
• A negative exponent means "one over": 10^-3 = 1 / 10^3 = 0.001

In Python, the exponent operator is ** (not ^, which means something else in code):

print(2 ** 3)      # 8
print(10 ** -3)    # 0.001

The rules you will use most

These rules only combine exponents when the base is the same. 2^3 × 3^2 cannot be merged into one power; you just compute 8 × 9 = 72.

Scientific notation

A number in scientific notation has two parts:

coefficient × 10^exponent, where the coefficient is at least 1 and less than 10.

• 4,500,000 = 4.5 × 10^6 (the decimal point moved 6 places left)
• 0.00032 = 3.2 × 10^-4 (the decimal point moved 4 places right)

A quick check: big numbers get positive exponents; numbers between 0 and 1 get negative exponents.

Computers and calculators print the same thing with an e: 4.5e6 means 4.5 × 10^6, and 3.2e-4 means 3.2 × 10^-4. You will see this in spreadsheets, Python and lab instruments.

print(4.5e6)              # 4500000.0
print(f"{0.00032:.1e}")   # 3.2e-04

Worked example 1: multiplying

(3 × 10^8) × (2 × 10^-3)

1. Multiply the coefficients: 3 × 2 = 6.
2. Add the exponents: 8 + (−3) = 5.
3. Answer: 6 × 10^5.

Worked example 2: re-normalising

(5 × 10^3) × (4 × 10^2)

1. Coefficients: 5 × 4 = 20.
2. Exponents: 3 + 2 = 5, giving 20 × 10^5.
3. 20 is not between 1 and 10, so rewrite 20 as 2.0 × 10^1 and add that exponent: 2.0 × 10^6.

Worked example 3: dividing

How many 2 × 10^-6 m bacteria fit end to end across 1 × 10^-2 m (one centimetre)?

(1 × 10^-2) ÷ (2 × 10^-6) = 0.5 × 10^4 = 5 × 10^3, so about 5,000.

Orders of magnitude

The exponent alone tells you the "size class" of a number. 3 × 10^5 and 8 × 10^5 are the same order of magnitude; 3 × 10^8 is three orders larger (about 1,000 times). Engineers use this for quick sanity checks: if your answer is off by an order of magnitude, look for a units or decimal-point slip.

Common mistakes

• Moving the decimal the wrong way. 0.0045 is 4.5 × 10^-3, not 4.5 × 10^3.
• Adding exponents when adding numbers. 10^3 + 10^3 = 2 × 10^3, not 10^6. Only multiplication adds exponents.
• Forgetting to re-normalise a coefficient of 10 or more (example 2).
• Typing ^ in code. In Python and JavaScript ^ is bitwise XOR; use **.

Quick recap

• Exponents count repeated multiplication; negative exponents mean division.
• Multiply: add exponents. Divide: subtract. Power of a power: multiply.
• Scientific notation keeps the coefficient between 1 and 10; e notation is the same thing for computers.

Try it

In the Do tab, finish a function that multiplies two numbers in scientific notation. The re-normalising step is already written for you.