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\(10^{\color{#a78bfa}{n}}\)

Scientific
Notation

Learn to express very large and very small numbers using powers of \(10\): a compact language used across science and engineering.

Pre-Algebra

Why Do We Need Scientific Notation?

Some numbers in the real world are enormous or microscopic. Writing every single zero is slow, error-prone, and hard to read.

Distance to the Sun
\(149{,}600{,}000{,}000\) meters: that is 12 digits long!
Width of a Virus
\(0.000000120\) meters: seven leading zeros are easy to miscount.

Review: Powers of \(10\)

A power of 10 tells you how many times to multiply (or divide) by \(10\). The exponent counts the number of places the decimal point moves.

Positive Exponents (bigger)
\(10^{0}\)
= 1
\(10^{1}\)
= 10
\(10^{2}\)
= 100
\(10^{3}\)
= 1,000
\(10^{6}\)
= 1,000,000
 Larger Start Negative Exponents (smaller)
\(10^{-1}\)
= 0.1
\(10^{-2}\)
= 0.01
\(10^{-3}\)
= 0.001
\(10^{-6}\)
= 0.000001
 Smaller

What Is Scientific Notation?

Scientific notation is a way to write any number as a product of two parts:

\(a\) × \(10\)\(^{n}\)
coefficient
\(1 \leq |a| < 10\)
power of 10
\(n\) is an integer

The Two Rules

1
Coefficient rule: \(a\) must satisfy \(1 \leq |a| < 10\). It has exactly one non-zero digit before the decimal point.
2
Exponent rule: \(n\) tells you how many places the decimal point moved. Positive \(n\) means the original number is large; negative \(n\) means it is small.

Positive vs. Negative Exponent

Positive Exponent
The original number is \(\geq 10\). Decimal moves right when converting back to standard form.

Example: \(3.2 \times 10^{4} = 32{,}000\)
Negative Exponent
The original number is < 1. Decimal moves left when converting back to standard form.

Example: \(7.1 \times 10^{-3} = 0.0071\)

How to Convert to Scientific Notation

1
Place the decimal after the first non-zero digit to form the coefficient \(a\).
2
Count the moves: how many places did you shift the decimal? That count becomes \(|n|\).
3
Choose the sign: if the original number is \(\geq 10\), \(n\) is positive. If it is < \(1\), \(n\) is negative.
4
Write the result: \(a \times 10^{n}\). Verify \(1 \leq a < 10\).

Example 1: Large Number

Convert \(93{,}000{,}000\) (the distance from Earth to the Sun in miles) to scientific notation.

9 3 , 0 0 0 , 0 0 0 . 7 places left
\(9.3 \times 10^{7}\)

Example 2: Small Number

Convert \(0.000052\) (the diameter of a human hair in meters) to scientific notation.

0 . 0 0 0 0 5 2 5 places right
\(5.2 \times 10^{-5}\)

Converting Back to Standard Form

To convert from scientific notation to standard form, reverse the process: move the decimal point by \(|n|\) places in the direction the exponent indicates.

1
Positive \(n\): move the decimal right \(n\) places, adding zeros as needed.
2
Negative \(n\): move the decimal left \(|n|\) places, adding leading zeros.

Example 3: Two-Step Conversion

A bacterium weighs about \(9.5 \times 10^{-13}\) grams. Write this in standard form.

1
Start with the coefficient: \(9.5\)
2
Exponent is \(-13\), so move decimal 13 places left.
3
Fill in 12 zeros after the decimal: \(0.00000000000095\) grams.

Comparing Numbers in Scientific Notation

To compare two numbers in scientific notation:

1
Compare exponents first. The number with the larger exponent is larger (for positive numbers).
2
If exponents are equal, compare the coefficients. The larger coefficient means the larger number.

Multiplying in Scientific Notation

To multiply two numbers in scientific notation, multiply the coefficients and add the exponents:

\((\color{#4ade80}{a} \times 10^{\color{#818cf8}{m}}) \times (\color{#4ade80}{b} \times 10^{\color{#818cf8}{n}}) = (\color{#fbbf24}{a \cdot b}) \times 10^{\color{#a78bfa}{m+n}}\)

Dividing in Scientific Notation

To divide, divide the coefficients and subtract the exponents:

\((\color{#4ade80}{a} \times 10^{\color{#818cf8}{m}}) \div (\color{#4ade80}{b} \times 10^{\color{#818cf8}{n}}) = (\color{#fbbf24}{a \div b}) \times 10^{\color{#a78bfa}{m-n}}\)

Common Mistakes

✕ Wrong Coefficient
Writing \(45 \times 10^{3}\) instead of \(4.5 \times 10^{4}\). The coefficient must be between 1 and 10.
✕ Wrong Sign
Using a positive exponent for a small number: \(0.003 \neq 3 \times 10^{3}\). Small numbers need a negative exponent.
✕ Miscounting Zeros
Off-by-one errors when counting decimal places. Always count the jumps, not the digits themselves.

Summary Table

Concept Standard Form Scientific Notation
Speed of light \(300{,}000{,}000\) m/s \(3 \times 10^{8}\)
One millimeter \(0.001\) m \(1 \times 10^{-3}\)
Multiply Multiply both parts Multiply \(a\)'s, add \(n\)'s
Divide Divide both parts Divide \(a\)'s, subtract \(n\)'s
Compare Count digits Compare \(n\) first, then \(a\)

Question 1 of 5

Which of the following correctly represents \(72{,}000\) in scientific notation?

Question 2 of 5

What is \(4.07 \times 10^{-5}\) written in standard form?

Question 3 of 5

Which number is larger: \(6.1 \times 10^{7}\) or \(8.9 \times 10^{5}\)?

Question 4 of 5

Simplify: \((3 \times 10^{5}) \times (4 \times 10^{3})\)

Question 5 of 5

A red blood cell has a diameter of about \(0.000007\) meters. Express this in scientific notation.