Unit 2: Order of Operations
To avoid making errors, mathematicians follow a series of steps to simplify expressions that use the four basic operations, grouping symbols, and exponents. We call these steps the "order of operations". In this unit, we explore how to use exponents and grouping symbols to perform mathematical operations in the correct order.
For example, we can use exponents to write out a repeat multiplication, such as 2 × 2 × 2 × 2 × 2, or 25. We also examine the concept of the greatest common factor (GCF), the largest positive integer that divides evenly into a given group of numbers with zero remainder. For example, for the set of numbers 18, 30 and 42, the GCF is 6. We also use the least common multiple (LCM) to determine the smallest number that two numbers can both divide into. For example, for the set of numbers 2 and 6, the LCM is 6. Finally, we learn how to perform mathematical operations with negative numbers.
Completing this unit should take you approximately 7 hours.
2.1: Greatest Common Factor
The greatest common factor (GCF) is the largest positive integer that divides evenly into a given group of numbers. For example, let's consider the numbers 12 and eight. We can divide both of those numbers by two: 12/2 = 6 and 8/2 = 4. But, we can also divide both of those numbers by four: 12/4 = 3 and 8/4 = 2. We cannot divide both numbers by any other factor. Therefore, the GCF for these numbers is four.
2.2: Least Common Multiple
Multiples are the result of multiplying two whole numbers together. We can write out multiples for any given number. For example, consider some multiples of three: 3 × 1 = 3; 3 × 2 = 6, 3 × 3 = 9. The first three multiples of 3 are 3, 6, and 9. We can see that different numbers sometimes have the same multiples. For example, consider some multiples of 2: 2 × 1 = 2, 2 × 2 = 4; 2 × 3 = 6. Both 3 and 2 have 6 as a multiple.
The least common multiple (LCM) is the smallest (or least) multiple that is the same between two or more whole numbers. In our example above, the least common multiple of 3 and 2 is 6.
2.3: Negative Numbers
So far, we have only been dealing with positive numbers. However, negative numbers are also very common, and we need to be able to perform operations with them. One common example is temperature.
We could have a temperature of −50.0 degrees or we could have a temperature of +50.0 degrees. We use the concept of negative numbers to calculate the difference between these two temperatures.
2.4: Adding and Subtracting Negative Numbers
When adding and subtracting negative numbers, we need to follow different rules than when adding and subtracting positive numbers.
2.5: Multiplying and Dividing Integers with Different Signs
When we multiply and divide integers, we must pay attention to the signs of the integers.
- When we multiply or divide two positive numbers, the result is always a positive number.
- Likewise, when we multiply or divide two negative numbers, the result is always a positive number.
- However, when we multiply or divide numbers with different signs (one positive and one negative), the result is a negative number.
Exponents are a way to simplify writing a number multiplied by itself multiple times. We write exponents as superscripts. For example: 42 = 4 × 4 = 16 and 43 = 4 × 4 × 4 = 64.
2.7: Order of Operations
When completing a calculation involving different mathematical operations, how do we decide what to do first? In mathematics, we follow a set of rules called the Order of Operations that determines the order for performing different types of calculations in a large multi-step problem.