Ontario Tech acknowledges the lands and people of the Mississaugas of Scugog Island First Nation.

We are thankful to be welcome on these lands in friendship. The lands we are situated on are covered by the Williams Treaties and are the traditional territory of the Mississaugas, a branch of the greater Anishinaabeg Nation, including Algonquin, Ojibway, Odawa and Pottawatomi. These lands remain home to many Indigenous nations and peoples.

We acknowledge this land out of respect for the Indigenous nations who have cared for Turtle Island, also called North America, from before the arrival of settler peoples until this day. Most importantly, we acknowledge that the history of these lands has been tainted by poor treatment and a lack of friendship with the First Nations who call them home.

This history is something we are all affected by because we are all treaty people in Canada. We all have a shared history to reflect on, and each of us is affected by this history in different ways. Our past defines our present, but if we move forward as friends and allies, then it does not have to define our future.

# Scientific Notation

## Introduction

Scientific notation is a useful way to deal with numbers that are extremely large or extremely small. Scientific notation is written as a product of some number between 1 and 10 and an integer power of 10. Scientific notation is of the form:

(Decimal between 1 and 10) x 10 integer exponent

CONVERTING TO SCIENTIFIC NOTATION

• Place the decimal between the 1st and 2nd digit or to right of the 1st non-zero number.
• Count the number of places the decimal was moved to reach the position in step 1.
• The number you got in step 2 becomes the integer exponent (negative if the decimal was moved to the right).

Example:

Express 0.0000000365 in scientific notation.

Solution:

Step 1: The decimal is placed between the 3 and 6.

Step 2: Since the decimal is moved 8 places to the right to reach this position, the exponent is -8.

Answer: $$3.65 \times 10^{-8}$$

Example:

Express 2390000 in scientific notation.

Solution:

Step 1: The decimal is placed between the 2 and 3.

Step 2: Since the decimal is moved 6 places to the left to reach this position, the exponent is 6.

Answer: $$2.39 \times 10^6$$

## CONVERTING TO STANDARD NOTATION

Move the decimal the number of places indicated by the exponent.

• If the exponent is positive, the decimal moves to the right.
• If the exponent is negative, the decimal moves to the left.

Often, zeros will be needed as placeholders in these situations.

Example: Express $$4.53 \times 10^{-4}$$ in standard notation.

Solution:

The decimal moves 4 positions to the left (since the exponent is - 4).

## ADDING AND SUBTRACTING IN SCIENTIFIC NOTATION

• Numbers must be written with the same exponent. Move the decimal place if needed to accomplish this (add 1 to the exponent for each time the decimal is moved to the left; subtract 1 from the exponent each time the decimal is moved to the right).
• Once the numbers have the same exponent, simply add or subtract them (the exponent stays the same).

Example: Find $$6.21 \times 10^4 + 1.33 \times 10^4$$

Solution: Since the numbers have the same exponent when written in scientific notation, we can simply add them to obtain:

$$\displaylines{6.21 \times 10^4 + 1.33 \times 10^4 \\ = (6.21 + 1.33) \times 10^4 \\ = 7.54 \times 10^4}$$

## MULTIPLYING AND DIVIDING IN SCIENTIFIC NOTATION

• When multiplying, the numbers are multiplied and the exponents are added.
• When dividing, the numbers are divided and the exponents are subtracted.

Example: Find $$8.6 \times 10^7 \div 2.0 \times 10^4$$.

Solution: We are dividing, so the numbers are divided and the exponents are subtracted:

$$\displaylines{8.6 \times 10^7 \div 2.0 \times 10^4 \\ =(\frac{8.6}{2.0}) \times 10^{(7-4)} \\ = 4.3 \times 10^3}$$

Scientific Notation Example 1:

Scientific Notation Example 2 - Converting to Standard Notation:

Scientific Notation Example 3 - Addition:

Scientific Notation Example 4 - Multiplication:

Scientific Notation Example 5: