Real Numbers

Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative. The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line.

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram. The ability to work comfortably with negative numbers is essential to success in algebra. For this reason we will do a quick review of adding, subtracting, multiplying and dividing integers. Integers are all the positive whole numbers, zero, and their opposites (negatives). As this is intended to be a review of integers, the descriptions and examples will not be as detailed as a normal lesson.

Adding and Subtracting Real Numbers

When adding integers we have two cases to consider. The first case is whether the signs match (both positive or both negative). If the signs match, we will add the numbers together and keep the sign. If the signs don’t match (one positive and one negative number) we will subtract the numbers (as if they were all positive) and then use the sign from the larger number. This means if the larger number is positive, the answer is positive. If the larger number is negative, the answer is negative.   Another way to think about subtracting is to think about the distance between the two numbers on the number line. In the example below, [latex]382[/latex] is to the right of 0 by [latex]382[/latex] units, and [latex]−93[/latex] is to the left of 0 by 93 units. The distance between them is the sum of their distances to 0: [latex]382+93[/latex]. The following video explains how to subtract two signed integers. https://youtu.be/ciuIKFCtWWU In the following video you will see an example of how to add three fractions with a common denominator that have different signs. https://youtu.be/P972VVbR98k In the following video are examples of adding and subtracting decimals with different signs. https://youtu.be/3FHZQ5iKcpI

Multiplying and Dividing Real Numbers

Multiplication and division are inverse operations, just as addition and subtraction are. You may recall that when you divide fractions, you multiply by the reciprocal. Inverse operations "undo" each other.

Multiply Real Numbers

Multiplying real numbers is not that different from multiplying whole numbers and positive fractions. However, you haven't learned what effect a negative sign has on the product. With whole numbers, you can think of multiplication as repeated addition. Using the number line, you can make multiple jumps of a given size. For example, the following picture shows the product [latex]3\cdot4[/latex] as 3 jumps of 4 units each. So to multiply [latex]3(−4)[/latex], you can face left (toward the negative side) and make three “jumps” forward (in a negative direction). The product of a positive number and a negative number (or a negative and a positive) is negative. The following video contains examples of how to multiply decimal numbers with different signs. https://youtu.be/7gY0S3LUUyQ The following video shows examples of multiplying two signed fractions, including simplification of the answer. https://youtu.be/yUdJ46pTblo To summarize:

• positive [latex]\cdot[/latex] positive: The product is positive.
• negative [latex]\cdot[/latex] negative: The product is positive.
• negative [latex]\cdot[/latex] positive: The product is negative.
• positive [latex]\cdot[/latex] negative: The product is negative.

You can see that the product of two negative numbers is a positive number. So, if you are multiplying more than two numbers, you can count the number of negative factors. The following video contains examples of multiplying more than two signed integers. https://youtu.be/rx8F9SPd0HE

Divide Real Numbers

You may remember that when you divided fractions, you multiplied by the reciprocal. Reciprocal is another name for the multiplicative inverse (just as opposite is another name for additive inverse). An easy way to find the multiplicative inverse is to just “flip” the numerator and denominator as you did to find the reciprocal. Here are some examples:

• The reciprocal of [latex]\frac{4}{9}[/latex] is [latex] \frac{9}{4}[/latex]because [latex]\frac{4}{9}\left(\frac{9}{4}\right)=\frac{36}{36}=1[/latex].
• The reciprocal of 3 is [latex]\frac{1}{3}[/latex] because [latex]\frac{3}{1}\left(\frac{1}{3}\right)=\frac{3}{3}=1[/latex].
• The reciprocal of [latex]-\frac{5}{6}[/latex] is [latex]\frac{-6}{5}[/latex] because [latex]-\frac{5}{6}\left( -\frac{6}{5} \right)=\frac{30}{30}=1[/latex].
• The reciprocal of 1 is 1 as [latex]1(1)=1[/latex].

When you divided by positive fractions, you learned to multiply by the reciprocal. You also do this to divide real numbers. Think about dividing a bag of 26 marbles into two smaller bags with the same number of marbles in each. You can also say each smaller bag has one half of the marbles.

[latex] 26\div 2=26\left( \frac{1}{2} \right)=13[/latex]

Notice that 2 and [latex] \frac{1}{2}[/latex] are reciprocals. Try again, dividing a bag of 36 marbles into smaller bags.

Number of bags	Dividing by number of bags	Multiplying by reciprocal
3	[latex]\frac{36}{3}=12[/latex]	[latex] 36\left( \frac{1}{3} \right)=\frac{36}{3}=\frac{12(3)}{3}=12[/latex]
4	[latex]\frac{36}{4}=9[/latex]	[latex]36\left(\frac{1}{4}\right)=\frac{36}{4}=\frac{9\left(4\right)}{4}=9[/latex]
6	[latex]\frac{36}{6}=6[/latex]	[latex]36\left(\frac{1}{6}\right)=\frac{36}{6}=\frac{6\left(6\right)}{6}=6[/latex]

Dividing by a number is the same as multiplying by its reciprocal. (That is, you use the reciprocal of the divisor, the second number in the division problem.) Now let's see what this means when one or more of the numbers is negative. A number and its reciprocal have the same sign. Since division is rewritten as multiplication using the reciprocal of the divisor, and taking the reciprocal doesn’t change any of the signs, division follows the same rules as multiplication. The following video explains how to divide signed fractions. https://youtu.be/OPHdadhDJoI

Properties of Real Numbers

[latex]a+b=b+a[/latex]

[latex]\left(-2\right)+7=5\text{ and }7+\left(-2\right)=5[/latex]

[latex]a\cdot b=b\cdot a[/latex]

[latex]\left(-11\right)\cdot\left(-4\right)=44\text{ and }\left(-4\right)\cdot\left(-11\right)=44[/latex]

[latex]a\left(bc\right)=\left(ab\right)c[/latex]

[latex]\left(3\cdot4\right)\cdot5=60\text{ and }3\cdot\left(4\cdot5\right)=60[/latex]

[latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]

[latex][15+\left(-9\right)]+23=29\text{ and }15+[\left(-9\right)+23]=29[/latex]

[latex]\begin{array}\text{ }8-\left(3-15\right) \hfill& \stackrel{?}{=}\left(8-3\right)-15 \\ 8-\left(-12\right) \hfill& =5-15 \\ 20 \hfill& \neq 20-10 \\ \text{ }\end{array}[/latex]

[latex]\begin{array}\text{ }64\div\left(8\div4\right)\hfill&\stackrel{?}{=}\left(64\div8\right)\div4 \\ 64\div2 \hfill& \stackrel{?}{=}8\div4 \\ 32 \hfill& \neq 2\end{array}[/latex]

[latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]

[latex]\begin{array}{ccc}\hfill 6+\left(3\cdot 5\right)& \stackrel{?}{=}& \left(6+3\right)\cdot \left(6+5\right) \\ \hfill 6+\left(15\right)& \stackrel{?}{=}& \left(9\right)\cdot \left(11\right)\hfill \\ \hfill 21& \ne & \text{ }99\hfill \end{array}[/latex]

[latex]a-b=a+\left(-b\right)[/latex]

[latex]12+\left(-1\right)\cdot \left(5+3\right)[/latex]

[latex]\begin{array}12-\left(5+3\right) \hfill& =12+\left(-1\right)\cdot\left(5+3\right) \\ \hfill& =12+[\left(-1\right)\cdot5+\left(-1\right)\cdot3] \\ \hfill& =12+\left(-8\right) \\ \hfill& =4 \end{array}[/latex]

[latex]\begin{array}12-\left(5+3\right) \hfill& =12+\left(-5-3\right) \\ \hfill& =12+\left(-8\right) \\ \hfill& =4\end{array}[/latex]

[latex]a+0=a[/latex]

[latex]a\cdot 1=a[/latex]

[latex]a+\left(-a\right)=0[/latex]

[latex]a\cdot \frac{1}{a}=1[/latex]

[latex]a\cdot \frac{1}{a}=\left(-\frac{2}{3}\right)\cdot \left(-\frac{3}{2}\right)=1[/latex]