Notes

Lesson 1.01

Real Numbers-A number that is rational or irrational. This is the highest set of numbers and includes all nonimaginary numbers.
Examples: −4, 3, √5, 3.2

Irrational: A number that cannot be expressed as a ratio of two numbers (fraction). This includes decimals that never stop.
Examples: square root of five, 3.2516…

Rational: A number that can be expressed as a ratio of two numbers (fraction). This includes decimals that terminate or repeat.
Examples:−14, 2.1, √4 (square root of 4 is 2, and is rational)

Integers: Positive and negative whole numbers including zeros. This category does not include any fractions or decimals. Examples: negative 2, 0, 4

Whole Numbers: All positive whole numbers including zero. This category does not include any negative whole numbers. Examples: 0, 3, 6

Natural Numbers: All positive whole numbers, not including zero. Examples: 4, 7, 9

A number line is shown with 5 tick marks to the left of 0 and 20 tick marks to the right of 0. The following terms appear as selectable buttons above: Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers. Natural Numbers has been selected for you.

When the term “Natural Numbers” is selected the following is revealed:


The numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 appear on the number line above, all to the right of 0.

When the term “Whole Numbers” is selected the following is revealed:


The numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 appear on the number line above.

When the term “Integers” is selected the following is revealed:


The numbers negative 5, negative 4, negative 3, negative 2, negative 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 appear on the number line above.

When the term “Rational Numbers” is selected the following is revealed:


The numbers negative 16 over 3, negative 4, negative 3.5, negative 2, negative 1.25, negative one-third, 0, one-half, 6, 11.1, 12.5, and 16 appear on the number line above. The number negative 16 over 3 appears just to the left of negative 5.

When the term “Irrational Numbers” is selected the following is revealed:


The numbers negative square root of 10, negative square root of 3, square root of 11, and square root of 20 appear on the number line above. The number negative square root of 10 appears just to the left of negative 3. The number negative square root of 3 appears just to the right of negative 2. The number square root of 11 appears just to the right of 3. The number square root of 20 appears about half way between 4 and 5.

•How are expressions rewritten in simplified form based on the mathematical operations in the expression?
•What is the correct order for performing mathematical operations in simplifying expressions?

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When adding and subtracting positive and negative numbers, you only have to ask yourself one question, same sign or different sign?
When you have same signs, think of the following phrase: Same signs, find the sum, keep the sign!

Step One: Same Signs
4 + 6 or -4 - 6
Step Two: Find the Sum
Add the absolute values of the numbers.
Covering up the signs leaves you with the absolute values.
Add 4 + 6.
Step Three: Keep the Sign
If both signs are positive, the answer is positive; if both signs are negative, the answer is negative.
So, 4 + 6 = 10, and –4 + –6 = –10.

Different Signs

When you have different signs, think of the following phrase: Different signs, find the difference, keep the sign of the larger number.

Step One: Different Signs
–4 + 5
Step Two: Find the Difference
Substract the absolute value of the numbers.
Remember, covering up the signs leaves you with the absolute values.
Subtract 5 – 4 = 1
After covering up the signs determine which is the larger number. The sign of this number will be the sign of your answer. Since 5 is larger, and is positive, the answer here is positive 1.
So, –4 + 5 = 1.

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When multiplying or dividing two numbers with same signs, the answer is positive. The opposite is true as well. When multiplying or dividing two numbers with different signs, the answer is negative.
There are two signs pushed together between the two terms and no number is attached to the first sign. If a term does not have a number, it is "understood" that there is an "invisible" 1. Thus, we can rewrite the problem as 4 – 1(–6).
There is a missing term outside the parenthesis again. It is understood that there is an "invisible" 1. Thus, we can rewrite the problem as 5 + 1(–2).

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Order of Operations

Parenthesis
Exponents
Multiplication/Division
Addition/Subtraction
Notice, there are two groups of operations, Multiplication/Division and Addition/Subtraction that are next to each other in the list. Keep in mind that you must do these operations that are on the same level from left to right.