Scientific Notation

Numbers with superpowers

Scientific Notation

Astronomers count stars by the billions. Biologists measure atoms by the trillionths. They all use the same shortcut: write any number as a small coefficient times a power of 10. Master the hops and huge numbers become easy.

Skill: scientific-notation
Chant it with me

“Count the hops! Big number, hop left, exponent’s positive. Small number, hop right, exponent’s negative.”

The exponent is just the number of places the decimal point jumps.

The form

Proper form

Every number becomes N × 10e where 1 ≤ N < 10 — one nonzero digit before the decimal point.

3.2 × 104 is proper. 32 × 103 is not (fix it: hop once more).

Big numbers → positive exponent

32,000: hop the decimal 4 places left → 3.2 × 104. The exponent counts your hops.

7,000,000 = 7 × 106 (6 hops left)

Small numbers → negative exponent

0.00045: hop the decimal 4 places right → 4.5 × 10−4. Tiny number, negative exponent.

0.0025 = 2.5 × 10−3 (3 hops right)

Multiplying

Multiply the N’s, add the exponents, then renormalize if N ≥ 10.

(6 × 104)(5 × 102) = 30 × 106 = 3 × 107

Worked examples

1Big number → scientific notation

Write 32,000 in scientific notation.

  1. Hop the decimal left until one nonzero digit remains: 32,000 → 3.2000. That is 4 hops.
  2. Coefficient N = 3.2, exponent = +4.

Answer: 3.2 × 104

2Small decimal → scientific notation

Write 0.00045 in scientific notation.

  1. Hop the decimal right until one nonzero digit remains: 0.00045 → 4.5. That is 4 hops.
  2. Small number → negative exponent. Coefficient N = 4.5, exponent = −4.

Answer: 4.5 × 10−4

3Scientific notation → standard form

Write 3.2 × 104 in standard form.

  1. Positive exponent 4 → hop the decimal 4 places right (that is ×10,000).
  2. 3.2 → 32 → 320 → 3,200 → 32,000.

Answer: 32,000

4Multiplying in scientific notation

Simplify (6 × 104)(5 × 102).

  1. Multiply coefficients: 6 × 5 = 30. Add exponents: 4 + 2 = 6 → 30 × 106.
  2. 30 is not proper (N must be < 10). Hop once: 30 × 106 = 3 × 107.

Answer: 3 × 107

Powers of 10 worth knowing

103
thousand
106
million
109
billion
1012
trillion
10−3
thousandth
10−6
millionth

Comparing? The bigger exponent almost always wins — 2 × 106 beats 3 × 105 even though 3 > 2. Only compare coefficients when exponents tie.

Quick check

1. Write 7,000,000 in scientific notation.

2. Write 2.5 × 10−3 in standard form.

3. Simplify (4 × 103)(2 × 105).

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