SL 1.1: Scientific Notation

Scientific notation is a way to express very large or very small numbers in a compact form. It involves writing numbers as a product of two parts: a coefficient and a power of ten. The general format is:

$a \times 10^n$

where:

• $a$ is a number greater than or equal to 1 and less than 10 (the coefficient).
• $n$ is an integer (the exponent).

Examples:

1. Large Numbers:
   • $5,000$ can be written as $5.0 \times 10^3$ (since $5,000 = 5 \times 1,000 = 5 \times 10^3$).
2. Small Numbers:
   • $0.00052$ can be written as $5.2 \times 10^{-4}$ (since $0.00052 = 5.2 \times 0.0001 = 5.2 \times 10^{-4}$).

Converting to Scientific Notation:

Converting a number to scientific notation involves a few straightforward steps. Let’s break it down with some examples.

Steps to Convert to Scientific Notation:

1. Identify the number: Start with the number you want to convert.

2. Move the decimal point:

   • Move the decimal point to the left to create a number between 1 and 10. Count how many places you move it. This count will be your exponent.
   • If you move it to the left, the exponent is positive. If you move it to the right, the exponent is negative.

3. Write in the format:

   • Write the number as $a\times 10^n$, where $a$ is your new number and $n$ is the exponent.

Examples:

1. Example 1: Converting 45,000

   • Move the decimal 4 places to the left: $4.5$
     
   • Exponent: $4$ (since we moved left).
   • Scientific notation: $4.5\times 10^4$
     

2. Example 2: Converting 0.00089

   • Move the decimal 4 places to the right: $8.9$
     
   • Exponent: $-4$ (since we moved right).
   • Scientific notation: $8.9\times 10^{-4}$
     

3. Example 3: Converting 320

   • Move the decimal 2 places to the left: $3.2$
     
   • Exponent: $2$
     
   • Scientific notation: $3.2\times 10^2$
     

4. Example 4: Converting 0.0000075

   • Move the decimal 7 places to the right: $7.5$
     
   • Exponent: $-7$
     
   • Scientific notation: $7.5\times 10^{-7}$
     

Practice:

Try converting these numbers to scientific notation:

1. $600000$
   
2. $0.00045$
   
3. $150$
   
4. $0.0000032$
   

Operations with Scientific Notation:

Performing operations with numbers in scientific notation can be simplified by following specific multiplication, division, addition, and subtraction rules. Here’s a guide on how to do each operation.

1. Multiplication

When multiplying two numbers in scientific notation:

$$(a\times 10^m)\times (b\times 10^n)=(a\times b)\times 10^{m+n}$$

Example: Multiply $(3.0\times 10^4)\times (2.0\times 10^3)$

$$=(3.0\times 2.0)\times 10^{4+3}=6.0\times 10^7$$

2. Division

When dividing two numbers in scientific notation:

$$\frac{a\times 10^m}{b\times 10^n}=\left(\frac{a}{b}\right)\times 10^{m-n}$$

Example: Divide $\frac{6.0\times 10^8}{3.0\times 10^2}$

$$=\left(\frac{6.0}{3.0}\right)\times 10^{8-2}=2.0\times 10^6$$

3. Addition and Subtraction

To add or subtract numbers in scientific notation, the exponents must be the same. If they are not, adjust one of the numbers so that the exponents match.

Example for Addition: Add $(2.5\times 10^3)+(4.0\times 10^2)$

1. Convert $4.0\times 10^2$ to match $10^3$:

   $$4.0\times 10^2=0.4\times 10^3$$
   

2. Now add:

   $$2.5\times 10^3+0.4\times 10^3=(2.5+0.4)\times 10^3=2.9\times 10^3$$
   

Example for Subtraction: Subtract $(5.0\times 10^6)-(2.0\times 10^5)$:

1. Convert $2.0\times 10^5$ to match $10^6$:

   $$2.0\times 10^5=0.2\times 10^6$$
   

2. Now subtract:

   $$5.0\times 10^6-0.2\times 10^6=(5.0-0.2)\times 10^6=4.8\times 10^6$$
   

Practice Problems

1. Multiply: $(4.5\times 10^2)\times (3.0\times 10^4)$
   
2. Divide: $\frac{8.0\times 10^5}{2.0\times 10^3}$
3. Add: $(1.2\times 10^3)+(3.0\times 10^2)$
   
4. Subtract: $(9.0\times 10^7)-(1.5\times 10^6)$