## Writing Numbers in Scientific Notation

Numbers that are very large (e.g. the speed of light, 299,792,458 m/s) or very small (e.g. the typical concentration of calcium in drinking water, 0.00165 M) are often easier to work with and compare when written in **scientific notation**, a type of exponential notation.

To correctly write a number in scientific notation:

- The number will be converted into an exponential form (see section below), with a digit term expressing the
*value*of the number, and an exponential term representing the*magnitude*of the number. - The digit term will only have ONE digit in front of the decimal. This digit cannot be a 0.
- Any trailing zeroes after the decimal can be kept in order to preserve precision / significant figures.
- It is correct to write numbers with any power of 10 in the exponential term, but normally numbers with moderate values (about $10^{-3}$ to $10^3$ ) will be written in decimal form.

Some examples of writing numbers in scientific notation:

Decimal form | Scientific Notation |

$ 124680 $ | $ 1.24680\times 10^5 $ |

$12468$ | $ 1.2468\times 10^4 $ |

$124.68$ | $ 1.2468\times 10^2 $ |

$1.2468$ | $ 1.2468\times 10^0 $ |

$0.0012468$ | $ 1.2468\times 10^{-3} $ |

$0.000124680$ | $ 1.24680\times 10^{-4} $ |

## Exponential Arithmetic (Scientific Notation)

Exponential notation is used to express very large and very small numbers as a product of two numbers. The first number of the product, the *digit term*, is usually a number not less than 1 and not equal to or greater than 10. The second number of the product, the *exponential term*, is written as 10 with an exponent. Some examples of exponential notation are:

$$1000=1×10^3$$ $$100=1×10^2$$ $$10=1×10^1$$ $$1=1×10^0$$ $$0.1=1×10^{−1}$$ $$0.001=1×10^{−3}$$ $$2386=2.386×1000=2.386×10^3$$ $$0.123=1.23×0.1=1.23×10^{−1}$$

The power (exponent) of 10 is equal to the number of places the decimal is shifted to give the digit number. The exponential method is particularly useful notation for very large and very small numbers. For example, $1,230,000,000 = 1.23×10^9$, and $0.00000000036 = 3.6×10^{−10}$.

#### Addition of Exponentials

Convert all numbers to the same power of 10, add the digit terms of the numbers, and if appropriate, convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

### Adding Exponentials Example

Add 5.00 × 10^{−5} and 3.00 × 10^{−3}.

**Solution**

$$3.00×10^{−3}=300×10^{−5}$$ $$(5.00×10^{−5})+(300×10^{−5})=305×10^{−5}=3.05×10^{−3}$$

#### Subtraction of Exponentials

Convert all numbers to the same power of 10, take the difference of the digit terms, and if appropriate, convert the digit term back to a number between 1 and 10 by adjusting the exponential term.

### Subtracting Exponentials Example

Subtract $4.0 \times 10^{−7}$ from $5.0\times 10^{−6}$.

**Solution**

$$4.0×10^{−7}=0.40×10^{−6}$$ $$(5.0×10^{−6})−(0.40×10^{−6})=4.6×10^{−6}$$

#### Multiplication of Exponentials

Multiply the digit terms in the usual way and add the exponents of the exponential terms.

### Multiplying Exponentials Example

Multiply $4.2 \times 10^{−8}$ by $2.0\times 10^{3}$.

**Solution**

$$(4.2×10^{−8})×(2.0×10^3)=(4.2×2.0)×10^{(−8)+(+3)}=8.4×10^{−5}$$

#### Division of Exponentials

Divide the digit term of the numerator by the digit term of the denominator and subtract the exponents of the exponential terms.

### Dividing Exponentials Example

Divide $3.6 \times 10^{−5}$ by $6.0\times 10^{-4}$.

**Solution**

$$\frac{3.6×10^{−5}}{6.0×10^{−4}}=(\frac{3.6}{6.0})×10^{(−5)−(−4)}=0.60×10^{−1}=6.0×10^{−2}$$

#### Squaring of Exponentials

Square the digit term in the usual way and multiply the exponent of the exponential term by 2.

### Squaring Exponentials Example

Square the number $4.0 \times 10^{−6}$.

**Solution**

$$(4.0×10^{−6})^2=4×4×10^{2×(−6)}=16×10^{−12}=1.6×10^{−11}$$

#### Cubing of Exponentials

Cube the digit term in the usual way and multiply the exponent of the exponential term by 3.

### Cubing Exponentials Example

Cube the number $2 \times 10^{4}$.

**Solution**

$$(2×10^4)^3=2×2×2×10^{3×4}=8×10^{12}$$

#### General rule: Exponentials Raised to a Power

From the previous two examples, you can deduce the general rule for exponentials raised to a power: that power is multiplied into the exponents of the exponential:

$$ (3.3 \times 10^3)^n\; =\; 3.3 \times 10^{(3n)} $$

#### Taking Square Roots of Exponentials

If necessary, decrease or increase the exponential term so that the power of 10 is evenly divisible by 2. Extract the square root of the digit term and divide the exponential term by 2.

### Finding the Square Root of Exponentials Example

Find the square root of $1.6 \times 10^{-7}$.

**Solution**

$$1.6×10^{−7}=16×10^{−8}$$ $$\sqrt{16×10^{−8}}=\sqrt{16}×\sqrt{10^{−8}}=\sqrt{16}×10^{−\frac{8}{2}}=4.0×10^{−4}$$