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Row 1 = Decimal Nos		Row 2 = Binary Nos		Row 3 = Hex. Nos		Row 5 = Hex. Label
00	01	02	03	04	05	06	07
000000	000001	000010	000011	000100	000101	000110	000111
02	24	07	19	15	36	46	11
Earth	Bounce	Army	Advance	Appropriate Action	Separate in Harmony	Survival	Peace
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08	09	10	11	12	13	14	15
001000	001001	001010	001011	001100	001101	001110	001111
16	51	40	54	62	55	32	34
Keen	Excite	Work	Lead	Femego	Inhale	Rythmn	Force
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16	17	18	19	20	21	22	23
010000	010001	010010	010011	010100	010101	010110	010111
08	03	29	60	39	63	48	05
Comrade	Difficult	Water	Bamboo	Inhibit	Subjective	Well	Hesitate
24	25	26	27	28	29	30	31
011000	011001	011010	011011	011100	011101	011110	011111
45	17	47	58	31	63	28	43
Socialise	Follow	Exhaust	Joy	Influence	Reform	Malego	Resolve
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32	33	34	35	36	37	38	39
100000	100001	100010	100011	100100	100101	100110	100111
23	27	04	41	52	22	18	26
Effort	Caring	Fool	Sacrifice	Calm	Tactful	Heal	Pause
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40	41	42	43	44	45	46	47
101000	101001	101010	101011	101100	101101	101110	101111
35	21	64	38	56	30	50	14
f
Progress	Justice	Objective	Exhibit	Travel	Fire	Cook	Help
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48	49	50	51	52	53	54	55
110000	110001	110010	110011	110100	110101	110110	110111
20	42	59	61	53	37	57	09
f
Tower	Money	Exhale	Love	Loyalty	Family	Gentle	Play
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56	57	58	59	60	61	62	63
111000	111001	111010	111011	111100	111101	111110	111111
12	25	06	10	33	13	44	01
War	Naive	Meet in Harmony	Appropriate Feeling	Retreat	Friend	Feel	Heaven
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Number Systems, in mathematics, various notational systems that have been or are being used to represent the abstract quantities called numbers. A number system is defined by the base it uses, the base being the number of different symbols, or numerals, required by the system to represent any of the infinite series of numbers. Thus, the decimal system in universal use today (except for computer application) requires ten different symbols, or digits, to represent numbers and is therefore a base-10 system.Throughout history many different number systems have been used; in fact, any whole number greater than 1 can be used as a base. Some cultures have used systems based on the numbers 3, 4 or 5. The Babylonians used the sexagesimal system, based on the number 60, and the Romans used (for some purposes) the duodecimal system, based on the number 12. The Mayans used the vigesimal system, based on the number 20. The binary system, based on the number 2, was used by some tribes and, together with the system based on 16, is used today in computer systems.

Place Values The position of a symbol denotes the value of that symbol in terms of exponential values of the base. That is, in the decimal system, the quantity represented by any of the ten symbols used—0, 1, 2, 3, 4, 5, 6, 7, 8, and 9—depends on its position in the number. Thus, the number 3,098,323 is an abbreviation for (3 × 106) + (0 × 105) + (9 × 104) + (8 × 103) + (3 × 102) + (2 × 101) + (3 × 100, or 3 × 1). The first 3 (reading from right to left) represents 3 units; the second 3, 300 units; and the third 3, 3 million units.

Two digits—0, 1—suffice to represent a number in the binary system; 6 digits—0, 1, 2, 3, 4, 5—are needed to represent a number in the sexagesimal system; and 16 digits—0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (ten), B (eleven), C (twelve), … , F (fifteen)—are needed to represent a number in the hexadecimal system. The number 30155 in the sexagesimal system is the number (3 × 64) + (0 × 63) + (1 × 62) + (5 × 61) + (5 × 60) = 3959 in the decimal system; the number 2EF in the duodecimal system is the number (2 × 162) + (14 × 161) + (15 × 160) = 751 in the decimal system.To write a given base-10 number n as a base-b number, divide (in the decimal system) n by b, divide the quotient by b, the new quotient by b, and so on until the quotient 0 is obtained. The successive remainders are the digits in the base-b expression for n. For example, to express 3959 (base 10) in the base 6, one writes

from which we see that 395910 = 301556. (The base is frequently written in this way as a subscript of the number.) The larger the base, the more symbols are required, but fewer digits are needed to express a given number.

Binary System The binary system plays an important role in computer technology. The first 20 numbers in the binary notation are 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000, 10001, 10010, 10011, 10100. Any number can be expressed in the binary system by the sum of different powers of two. For example, starting from the right, 10101101 represents (1 × 20) + (0 × 21) + (1 × 22) + (1 × 23) + (0 × 24) + (1 × 25) + (0 × 26) + (1 × 27) = 173.

Arithmetic operations in the binary system are extremely simple. The basic rules are: 1 + 1 = 10, and 1 × 1 = 1. Zero plays its usual role: 1 × 0 = 0, and 1 + 0 = 1. Addition, subtraction, and multiplication are done in a fashion similar to that of the decimal system:

Because only two digits (or bits) are involved, the binary system is used in computers, since any binary number can be represented by, for example, the positions of a series of on-off switches. The on position corresponds to a 1, and the off position to a 0. Instead of switches, magnetized dots on a magnetic tape or disk also can be used to represent binary numbers: a magnetized dot stands for the digit 1, and the absence of a magnetized dot is the digit 0. Flip-flops—electronic devices that can only carry two distinct voltages at their outputs and that can be switched from one state to the other state by an impulse—can also be used to represent binary numbers. Logic circuits in computers carry out the different arithmetic operations of binary numbers; the conversion of decimal numbers to binary numbers for processing, and of binary numbers to decimal numbers for the readout, is done electronically.

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