Algebra Chapter 1 Section 2: Operations and Properties of Real Numbers

Here, I will cover some of the basic operations and properties of real numbers. I say real numbers to differentiate from imaginary and complex numbers, both of which deal with the square root of -1, $\sqrt{-1}=i$. This course will not deal with the square root of -1, so all numbers will be real in this course. There are courses which deal with i, which typically have the word "Complex" in their name.

The Absolute Value

The absolute value (given by $\left | a \right |$) is the distance from the zero on the number line, regardless of the sign. This is to say $\left | -a |=\left | a |$.  The absolute value of -2 is 2, the absolute value of 2 is 2.  It does not matter what the initial value is, the sign after running it through the absolute value function will always, under every circumstance ever, be positive.

Inequalities

There are two types of inequality, less than and "greater than".  There are also two flavors of inequality, with and without the equal sign.  This means that there are four types of inequality: "less than" (<), "greater than" (>), "less than or equal to" (≤), and "greater than or equal to" (≥).  The only difference between "less than" and "less than or equal to" is that in the latter, there is a possibility of equality, even if it's not guaranteed, while in the former, there is no possibility of equality.  The same concept applies to "greater than" and "greater than or equal to".

Properties of Multiplication and Division

When we multiply two numbers of the same sign, the answer will always be positive.  When we multiply two numbers with opposite signs, then the answer will always be negative.  The same concept applies to division; one both numbers of a division are of the same sign, the answer is positive, while two different signs come up negative.  In general, if there are an even number of negative numbers (or no negative numbers) being multiplied together with any number of positive numbers, than the answer will be positive; if there is an odd number of negative numbers, though, the answer will be negative.

When there is a negative sign in a fraction, it does not matter where the negative sign is placed, for anywhere you could place it yields the same number.  For example, for two values a and b, $-\frac{a}{b}=\frac{-a}{b}=\frac{a}{-b}$.  There is no difference between those three fractions.

When constructing a fraction, be very careful to not divide by zero.  The reason you've probably heard that we cannot divide by zero is simple.  I'm going to use an example.  Let's say we have $\frac{5}{0}=7$.  If we multiply both sides by zero, thus "canceling the zeros" (I can already hear the yelling of mathematicians), we get $5=7 \times 0$, or 5=0.  Since we know that $5 \neq 0$, we know that this can not hold true.  In fact, this is only true if we have zero in the numerator as well as the denominator, but in that case, we can have $\frac{0}{0}=3$ as well as $\frac{0}{0}=10$, which means that 3=10, and we all know that $3 \neq 10$.  These are two parts of a proof by contradiction to show that we cannot divide by zero because inequalities suddenly become equal,
$1=25million=5672515410534017560169028749258751849215760154687045015471$.

Understatement of the millennium, Bill.

Commutation, Association, Distribution, and Equivalent Equations.

Two expressions are said to be equal when their solutions are the same.  Two equations are said to be equivalent when they have the same value for all possible replacements of all variables.  This holds true for the Law of Commutation, which states that, for any two values a and b, a+b=b+a and $a \times b=b \times a$.  This also holds true for the Law of Association, $a+(b+c)=(a+b)+c$ and $a \times (b \times c)=(a \times b) \times c$. It also holds true for the law of Distribution, $a \times (b+c)=a \times b + a \times c$.

That's the end of this section.  If you have any questions, please leave them is the comments.  Like and share this post if you found it helpful.  And until next time, stay curious.

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