# Lesson 8.1: Real Numbers

### The Multiplication Dot

Another way that multiplication is shown in print is with the multiplication dot. In braille, the multiplication dot uses one cell, dots one six, which is the second cell of the symbol that is used for the multiplication cross. In fact, the hollow dot is another modification of the dots one six, preceding it with dots four six in the first cell. Since the multiplication dot is never used in spatial arrangements, it is only necessary to learn how to use it in horizontal mathematical expressions.

As with any sign of operation, when brailling a multiplication dot, do not space between the multiplication dot and the symbols with which it is used. The guidelines and rules for using this symbol are exactly the same as those for the multiplication cross. Here are some examples of simple multiplication using the multiplication dot.

Most texts do not display the multiplication dot followed by the decimal point because of the possibility of confusion for the print reader. This is particularly troublesome with handwritten material. Thus 3.7⋅0.2 would probably be displayed as (3.7)(0.2) using parentheses to indicate multiplication.

Just as with any other sign of operation, the multiplication dot can be used with other symbols.

### The Radical Sign

The most common use of the radical symbol, dots three four five, is to represent square roots, a root of the second degree. The radical sign is also used to indicate radicals of a higher degree, such as cubed or fourth roots; these will be presented later. The terms square root, root, and radical are used interchangeably.

The radical symbol is paired with the termination indicator, dots one two four five six, when it is followed by a value. When it stands alone, the termination indicator is not used. As the name indicates (implies), termination indicator is strictly a symbol used as a convention in braille and has no corresponding symbol in print. When paired with the radical sign, the termination indicator is used to mark the end of the radicand, the value following the radical sign. In print, a horizontal line extends from the radical sign and over the values, expressions, variables, or abbreviations which comprise the radicand. This line is called the vinculum and serves the same purpose as the termination indicator.

A radical is to be brailled in a horizontal manner except in two circumstances. It is displayed vertically if it is represented in a spatial arrangement. It is displayed vertically if the root is displayed above the vinculum when showing the process of extracting a root. The spatial procedures used with long division are applied in a similar manner to root extractions and the termination symbol is not used.

### Example 1

five times two times ten using the multiplication dot

⠼⠢⠡⠆⠡⠂⠴

### Example 2

six times two equals twelve using the multiplication dot

⠼⠖⠡⠆⠀⠨⠅⠀⠼⠂⠆

### Example 3

three point seven times zero point two using the multiplication dot

⠼⠒⠨⠶⠡⠴⠨⠆

### Example 4

one half times two thirds using the multiplication dot

⠹⠂⠌⠆⠼⠡⠹⠆⠌⠒⠼

### Example 5

two point five times three point two using parentheses only

⠷⠆⠨⠢⠾⠷⠒⠨⠆⠾

### Example 6

the square root of four

⠜⠲⠻

### Example 7

the negative value of the square root of four

⠤⠜⠲⠻

### Example 8

the square root of the expression three plus six

⠜⠒⠬⠖⠻

### Example 9

five times the square root of three

⠼⠢⠜⠒⠻

### Example 10

seven times the square root of three minus two times the square root of three

⠼⠶⠜⠒⠻⠤⠆⠜⠒⠻

### Example 11

eight times the square root of two divided by four times the square root of two using a division symbol

⠼⠦⠜⠆⠻⠨⠌⠲⠜⠆⠻

### Example 12

eight time the square root of two over four times the square root of two written as a fraction

⠹⠦⠜⠆⠻⠌⠲⠜⠆⠻⠼

### Example 13

the square root of the fraction twelve over three

⠜⠹⠂⠆⠌⠒⠼⠻

### Example 14

the square root of two multiplication dot square root of three

⠜⠆⠻⠡⠜⠒⠻