Numeral verses Number

lA numeral is a symbol used to represent some value.

lA number is the value that the symbol or a group of symbols represent.

lIn order to have a number system, it is necessary to define numerals (symbols) to represent key values.

Evolution of a Number System

lTypes of number systems in order of evolution.

–Tally

–Grouping

–Multiplicative grouping

–Positional

Evolution of a Number System

lIn a tally system, each stroke or tally mark represents one.

lIn a grouping system, there is a symbol for select numbers; numbers are formed by grouping and repeating the symbols.

lIn a multiplicative grouping system, there is a symbol for each value 1 through 9 (the multipliers), and also for select other numbers (usually powers of 10 or some other common base).

lIn a positional system, there is a symbol for each number 0 through 9 (base ten, other bases will be discussed later).

Writing Numbers in a Tally System

lIn a tally system, there is one symbol and it represents the number “one.”

lTo represent large numbers in a tally system you need to repeat the symbol as many times as the value of the number.

lTo make tallies easier to read, they are often grouped in fives.

Writing Numbers in a Grouping System

lTo write a number in a grouping system you repeat the symbol representing the appropriate value(s) until you have the desired value.

lExample: To write the number 53 in a grouping system, you would need to write five copies of the symbol for “ten” and 3 copies of the symbol for “one.”

Writing Numbers in a Multiplicative Grouping System

lTo write a number in a multiplicative grouping system, you need a multiplier followed by the symbol representing the value of the appropriate power of ten (or base of choice).

lExample: To write the number 53 in a multiplicative grouping system you would have two “Groups” of two, the multiplier representing the number 5, followed by the symbol for 10 (5x10=50), then the second group would be the multiplier representing the number 3, followed by the symbol for one (3x1=3). (note: the symbol for one may be omitted in a multiplicative system.

Writing Numbers in a Positional System

lIn a positional system no multiplier is needed, the value of the symbol is understood by its position in the number.

lThe Hindu-Arabic System (the one we use) is a positional system.

lTo represent a number in a positional system you simply put the numeral in the appropriate place in the number, and its value is determined by its location.

Matthews’ Number System

lWe define the symbols as follows:

–1 " 10 ,

–2 " 100 1

–3 $ 1,000 6

–4 % 10,000 8

–5 & 100,000 <

–6 '

–7 (

–8 )

–9 *

Matthews’ number system is a multiplicative system.

Examples

lWrite the number 123,324 in Matthews’ number system.

–Solution: "< "8 $6 $1 ", %

lAdd: "< $8 "6 $1 ", " + $< "8 "6 "1 "

–Solution: &< &8$6 %1 ", $

lConvert the above numbers to Hindu-Arabic.

–Solution: 231,321 + 322,102 = 553,423