[Func-Num] 6.2 Operations, Properties, and Applications of Real Numbers

Date: 2023.01.13 Updated: 2023.01.14

Categories: Func-Num

📋 This is my note-taking from what I learned in the class “Math175-002 Functions & Number Systems”

Objectives

Operations of additions, substraction, multiplication, and division of signed numbers

The rules for order of operations

Properties of addition and multiplication of real numbers

Change in investment and meteorological data using subtraction and absolute value

Operations on Real Numbers

1. Definition of Subtraction

The result of subtracting two numbers is their difference.
“The difference of A and B” is interpreted as “A-B”
For all real numbers A and B, A-B = A+(-B) → Change the sign of the subtrahend and Add.
In A-B, A is the minuend, and B is the subtrahend.
What is Subtrahend?

2. Multiplying Real Numbers

The result of multiplying two numbers is their product.
The two numbers being multiplied are factors.

Like Signs:

Multiply two numbers with the same sign by multiplying their absolute values to find the absolute value of the product.

The product is positive.

Unlike Signs:

Multiply two numbers with different signs by multiplying their absolute values to find the absolute value of the product.

The product is negative.

Sign Rules for Multiplication

(+)*(+) = +

(-)*(-) = +

(+)*(-) = -

(-)*(+) = -

3. Dividing Real Numbers

The result of dividing two numbers is their quotient.
“The quotient of A and B” is interpreted as “A/B”
In the quotient A/B, where B≠0, A is the dividend (or numerator), and B is the divisor (or denominator).

Like Signs:

Divide two numbers with the same sign by dividing their absolute values to find the absolute value of the quotient.

The quotient is positive.

Unlike Signs:

Divide two numbers with different signs by dividing their absolute values to find the absolute value of the quotient.

The quotient is negative.

Sign Rules for Division

(+)/(+) = +

(-)/(-) = +

(+)/(-) = -

(-)/(+) = -

Division Involving Zero

A/0 is undefined for all A

0/A = 0 for all nonzero A

Order of Operations

If parentheses or square brackets are present:

Step 1 Work separately above and below any fraction bar.
Step 2 Use the rules below within each set of parentheses or square brackets. Start with the innermost set and work outward.

If no parentheses or brackets are present:

Step 1 Apply any exponents.
Step 2 Do any multiplications or divisions in the order in which they occur, working from left to right.
Step 3 Do any additions or subtractions in the order in which they occur, working from left to right.

“Please Excuse My Dear Aunt Sally”

Parenthese

Exponents

Multiply

Divide

Add

Subtract

→ M and D have equal priority → A and S have equal priority

Properties of Addition and Multiplication of Real Numbers

For real numbers a, b, and c, the following properties hold.

Closure Properties:

The sum of two real numbers and the product of two real numbers are themselves real numbers

a + b and a * b are real numbers.

Commutative Properties:

Two real numbers may be added or multiplied in either order without affecting the result

a + b = b + a

a * b = b * a

Associative Properties

Group terms or factors in any manner we wish without affecting the result

(a + b) + c = a + (b + c)

(a * b) * c = a * (b * c)

Identity Properties

The number 0 is the identity element for addition. Adding 0 to a real number will always yield that real number.
The number 1 is the identity element for multiplication. Multiplying a real num-ber by 1 will always yield that real number.
- There is a real number 0 such that → a + 0 = a and 0 + a = a.
- There is a real number 1 such that → a * 1 = a and 1 * a = a.

Inverse Properties

Each real number a has an additive inverse, -a, such that the sum of a and its additive inverse is the additive identity element 0.
Each nonzero real number a has a multiplicative inverse, or reciprocal, 1/a, such that the product of a and its multiplicative inverse is the multiplicative identity element 1.
- For each real number a, there is a single real number -a such that → a + (−a) = 0 and (−a) + a = 0.
- For each nonzero real number a, there is a single real number 1/a such that → a * 1/a = 1 and 1/a * a = 1.

Distributive Property of Multiplication with Respect to Addition

Express certain products as sums and certain sums as products

a * (b + c) = a * b + a * c

(b + c) * a = b * a + c * a

Exercise

Section 6.2 → No. 31 ~ 43 (odds)

-6
27
0
-1
-4
7
13

Seyeon