Easy Algebra Step-by-Step

Numbers of Algebra 5

nates, and

2

3

= 0. 666 ... is a rational number whose decimal representation

repeats. You can show a repeating decimal by placing a line over the block of

digits that repeats, like this:

2

3

= 06 .. 6 You also might fi nd it convenient to round

the repeating decimal to a certain number of deci-

mal places. For instance, rounded to two decimal

places,^2

3

≈ 06. 7.

The irrational numbers are the real numbers

whose decimal representations neither terminate nor repeat. These numbers

cannot be expressed as ratios of two integers. For instance, the positive num-

ber that multiplies by itself to give 2 is an irrational number called the posi-

tive square root of 2. You use the square root symbol () to show the positive

square root of 2 like this: 2. Every posi-

tive number has two square roots: a posi-

tive square root and a negative square root.

The other square root of 2 is − 2. It also

is an irrational number. (See Chapter 3 for

an additional discussion of square roots.)

You cannot express 2 as the ratio of two integers, nor can you express

it precisely in decimal form. Its decimal equivalent continues on and on with-

out a pattern of any kind, so no matter how far you go with decimal places,

you can only approximate 2. For instance,

rounded to three decimal places, 21. 414.

Do not be misled, however. Even though

you cannot determine an exact value for

2 , it is a number that occurs frequently in

the real world. For instance, designers and builders encounter 2 as the

length of the diagonal of a square that has sides with length of 1 unit, as

shown in Figure 1.6.

The symbol ≈ is used to mean “is

approximately equal to.”

The number 0 has only one square root,

namely, 0 (which is a rational number). The

square roots of negative numbers are not

real numbers.

Not all roots are irrational. For instance,

36 =6 and^3 − 64 =− 4 are rational

numbers.

Figure 1.6 Diagonal of unit square

1 unit

1 unit

√2 units