3.9: Summary of Key Concepts

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Summary of Key Concepts

Positive and Negative Numbers
A number is denoted as positive if it is directly preceded by a "+" sign or no sign at all. A number is denoted as negative if it is directly preceded by a "−" sign.

Opposites
Opposites are numbers that are the same distance from zero on the number line but have opposite signs.

Double-Negative Property
\(−(−a)=a\)

Absolute Value (Geometric)
The absolute value of a number a, denoted |a|, is the distance from a to 0 on the number line.

Absolute Value (Algebraic)
\(|a|=\left\{\begin{array}{cl}
a & \text { if } a \geq 0 \\
-a & \text { if } a<0
\end{array}\right.\)

Addition of Signed Numbers
To add two numbers with
like signs, add the absolute values of the numbers and associate the common sign with the sum.
unlike signs, subtract the smaller absolute value from the larger absolute value and associate the sign of the larger absolute value with the difference.

Addition with 0
0+any number=that particular number, that is, 0+a=a for any real number a.

Additive Identity
Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity.

Definition of Subtraction
\(a−b=a+(−b)\)

Subtraction of Signed Numbers
To perform the subtraction \(a−b\), add the opposite of \(b\) to \(a\), that is, change the sign of b and add.

Multiplication and Division of Signed Numbers
\(\begin{array}{l}
(+)(+)=+\dfrac{(+)}{(+)}=+\dfrac{(+)}{(-)}=- \\
(-)(-)=+ \\
(+)(-)=- \\
(-)(+)=-\dfrac{(-)}{(-)}=+\dfrac{(-)}{(+)}=-
\end{array}\)

Reciprocals
Two numbers are reciprocals of each other if their product is \(1\). The numbers \(4\) and \(\dfrac{1}{4}\) are reciprocals since \((4) (\dfrac{1}{4})=1\).

Negative Exponents
If \(n\) is any natural number and \(x\) is any nonzero real number, then \(x^{-n}=\dfrac{1}{x^n}\).

Writing a Number in Scientific Notation
To write a number in scientific notation:

Move the decimal point so that there is one nonzero digit to its left.
Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

Converting from Scientific Notation:
positive exponent
To convert a number written in scientific notation to a number in standard form when there is a positive exponent as the power of 10, move the decimal point to the right the number of places prescribed by the exponent on the 10.

Negative Exponent
To convert a number written in scientific notation to a number in standard form when there is a negative exponent as the power of 10, move the decimal point to the left the number of places prescribed by the exponent on the 10.