Scientific Notation

Scientific notation is a way to write very large or very small numbers more compactly. Instead of writing 93,000,000 miles (the distance from Earth to the Sun), scientists write $9.3 \times 10^7$ miles. Instead of writing 0.00000000016 meters (the size of a hydrogen atom), they write $1.6 \times 10^{-10}$ meters.

This system makes it easier to read, write, and calculate with extreme numbers. It's used throughout science, engineering, and any field dealing with very large or very small quantities.

The Format

A number in scientific notation has two parts:

$$a \times 10^n$$

where $a$ is a number greater than or equal to 1 but less than 10, and $n$ is an integer (the exponent on 10).

The $a$ is called the coefficient or significand. The $10^n$ part is what makes the number large or small.

Example: $3.2 \times 10^5$ means 3.2 times 100,000, which equals 320,000.

Example: $7.5 \times 10^{-3}$ means 7.5 times 0.001, which equals 0.0075.

Converting from Standard to Scientific Notation

To convert a large number to scientific notation, move the decimal point to the left until you have a number between 1 and 10. Count how many places you moved—that's your exponent.

Example: Convert 45,000 to scientific notation.

Move the decimal 4 places to the left: 45,000 becomes 4.5

The exponent is 4.

Answer: $4.5 \times 10^4$

Example: Convert 8,730,000,000 to scientific notation.

Move the decimal 9 places left: 8.73

Answer: $8.73 \times 10^9$

To convert a small number (between 0 and 1) to scientific notation, move the decimal to the right until you have a number between 1 and 10. Count how many places—that's your exponent, but it's negative.

Example: Convert 0.0056 to scientific notation.

Move the decimal 3 places right: 5.6

The exponent is -3.

Answer: $5.6 \times 10^{-3}$

Example: Convert 0.000000721 to scientific notation.

Move the decimal 7 places right: 7.21

Answer: $7.21 \times 10^{-7}$

Converting from Scientific to Standard Notation

To convert from scientific notation back to standard form, look at the exponent.

If the exponent is positive, move the decimal that many places to the right.

Example: $3.8 \times 10^4 = 38,000$

Move the decimal 4 places right.

Example: $1.25 \times 10^6 = 1,250,000$

If the exponent is negative, move the decimal that many places to the left.

Example: $6.7 \times 10^{-3} = 0.0067$

Move the decimal 3 places left.

Example: $9.2 \times 10^{-5} = 0.000092$

Multiplying Numbers in Scientific Notation

To multiply, multiply the coefficients and add the exponents.

Example: $(2 \times 10^3) \times (4 \times 10^5)$

Multiply coefficients: $2 \times 4 = 8$

Add exponents: $3 + 5 = 8$

Answer: $8 \times 10^8$

Example: $(3.5 \times 10^4) \times (2 \times 10^{-2})$

Multiply: $3.5 \times 2 = 7$

Add exponents: $4 + (-2) = 2$

Answer: $7 \times 10^2$

Sometimes the result isn't in proper scientific notation because the coefficient isn't between 1 and 10. In that case, adjust.

Example: $(6 \times 10^7) \times (5 \times 10^2)$

Multiply: $6 \times 5 = 30$

Add exponents: $7 + 2 = 9$

We get $30 \times 10^9$, but 30 isn't between 1 and 10.

Convert 30 to scientific notation: $3.0 \times 10^1$

Combine: $3.0 \times 10^1 \times 10^9 = 3.0 \times 10^{10}$

Dividing Numbers in Scientific Notation

To divide, divide the coefficients and subtract the exponents.

Example: $\frac{8 \times 10^6}{2 \times 10^3}$

Divide: $\frac{8}{2} = 4$

Subtract exponents: $6 - 3 = 3$

Answer: $4 \times 10^3$

Example: $\frac{9 \times 10^2}{3 \times 10^5}$

Divide: $\frac{9}{3} = 3$

Subtract: $2 - 5 = -3$

Answer: $3 \times 10^{-3}$

Adding and Subtracting

Adding and subtracting in scientific notation is trickier because the exponents must be the same before you can combine the coefficients.

Example: $3 \times 10^4 + 5 \times 10^4$

Same exponents, so just add coefficients: $$= (3 + 5) \times 10^4 = 8 \times 10^4$$

Example: $6 \times 10^5 - 2 \times 10^5$

$$= (6 - 2) \times 10^5 = 4 \times 10^5$$

When exponents are different, you need to adjust one number to match the other's exponent.

Example: $4 \times 10^6 + 7 \times 10^4$

Rewrite $4 \times 10^6$ with exponent 4: $$4 \times 10^6 = 400 \times 10^4$$

Now add: $$400 \times 10^4 + 7 \times 10^4 = 407 \times 10^4$$

Convert to proper form: $$= 4.07 \times 10^6$$

Real-World Examples

The speed of light is about $3 \times 10^8$ meters per second.

The mass of Earth is approximately $5.97 \times 10^{24}$ kilograms.

A single cell in your body is about $1 \times 10^{-5}$ meters across.

The charge of an electron is $-1.6 \times 10^{-19}$ coulombs.

These numbers would be unwieldy to write in standard form, which is why scientific notation is so valuable.

Work Through These

Write 250,000 in scientific notation.
Write 0.00034 in scientific notation.
Write $7.2 \times 10^{-4}$ in standard notation.
Multiply: $(3 \times 10^5) \times (4 \times 10^2)$
Divide: $\frac{8 \times 10^7}{2 \times 10^3}$
Add: $2 \times 10^3 + 5 \times 10^3$

Solutions:

$2.5 \times 10^5$
$3.4 \times 10^{-4}$
$0.00072$
$(3 \times 4) \times 10^{5+2} = 12 \times 10^7 = 1.2 \times 10^8$
$\frac{8}{2} \times 10^{7-3} = 4 \times 10^4$
$(2 + 5) \times 10^3 = 7 \times 10^3$

Easy Mistakes

Writing the coefficient outside the range of 1 to 10 means your answer isn't in proper scientific notation. If you end up with $25 \times 10^4$, you need to adjust it to $2.5 \times 10^5$. The first number must be at least 1 but less than 10.

Moving the decimal the wrong direction happens when you're not paying attention to the number's size. For numbers larger than 10, the decimal moves left (and you get a positive exponent). For numbers between 0 and 1, the decimal moves right (and you get a negative exponent).

Forgetting to add exponents when multiplying is easy to do if you're on autopilot. The rule is: multiply the coefficients, add the exponents. Both steps matter.

Confusing the subtraction direction when dividing causes wrong answers. It's always top exponent minus bottom exponent, not the other way around.

Not adjusting exponents before adding or subtracting makes the problem unsolvable. You can't directly combine $3 \times 10^4$ and $5 \times 10^6$ until you rewrite one of them so both have the same exponent.