Chapter 2: Number Bases

2.1 Number Bases and Place Values

• Number bases are number systems; base ten is the decimal system used widely in daily life. Bases 2, 8, 10 and 16 are used in computer science. (p.34)
• Digits are the symbols used to form a number. Each base uses digits from 0 up to one less than the base (e.g. base two uses only 0 and 1). (p.35)
• Notation: 325 read as “three two base five”; the base is written as a subscript. (p.35)
• Place values are repeated multiplications of the base: a⁰, a¹, a², …, aⁿ (a = base, n = power; a⁴ = a × a × a × a). (p.35)
• Digit value = digit × place value of its position. (p.36)
• Number value = sum of the digit values (collection process). Base two has only digits 0 and 1; e.g. 11001₂ = 16 + 8 + 0 + 0 + 1 = 25₁₀. (p.37)
• Writing the base sign for a base-ten number is optional. (p.38)
• Examples: 340₅ = 95; 341₇ = 176; 1506₈ = 838 (all in base ten). (p.39)
• Calculator check: MODE → BASE, choose the base (e.g. OCT), enter the number, then press DEC. (p.39)

2.1 Converting Between Bases

• A number can be converted using division by place value or division by the base value. (p.41)
• Base ten → another base (repeated division): divide by the base, record remainders, and read the digits from the bottom upwards. e.g. 563₁₀ = 4223₅. (p.41)
• 563₁₀ = 1063₈ (division by 8, remainders read from the bottom upwards). (p.42)
• Base p → base q: convert to base ten first, then to base q. e.g. 253₆ = 105₁₀ = 126₉. (p.42)
• 334₅ = 94₁₀ = 1011110₂. (p.43)
• Base two → base eight: each base-eight digit equals three base-two digits. Separate the digits into groups of three from right to left, find each group’s value, then combine. e.g. 110111₂ = 67₈. (p.43)
• 1101101₂ = 155₈ (pad the leftmost group with leading zeros: 001 101 101). (p.44)
• Base eight → base two: convert each octal digit to three binary digits and combine. e.g. 517₈ = 101001111₂; 725₈ = 111010101₂. (p.44)

Base 2, base 10 and base 16 (p.44)

Base 2	Base 10	Base 16
0000	0	0
0001	1	1
0010	2	2
0011	3	3
0100	4	4
0101	5	5
0110	6	6
0111	7	7
1000	8	8
1001	9	9
1010	10	A
1011	11	B
1100	12	C
1101	13	D
1110	14	E
1111	15	F

Base 2 and base 8 equivalents (p.43)

Base 2	000	001	010	011	100	101	110	111
Base 8	0	1	2	3	4	5	6	7

2.1 Addition and Subtraction in Various Bases

• Two methods: (a) vertical form, (b) convert to base ten, calculate, then convert the answer back to the required base. (p.45)
• Addition (vertical form): add digits from right to left; convert each column sum to the given base, write the units digit and carry the rest to the next place value. (p.46)
• 110₂ + 111₂ = 1101₂; 673₈ + 175₈ = 1070₈; 1837₉ + 765₉ = 2713₉. (p.46)
• Subtraction (vertical form): subtract digits from right to left; when borrowing, transfer according to the base (e.g. use 6 in base six). The difference is always less than the base. (p.47)
• 4005₆ − 325₆ = 3240₆. (p.47)
• 6241₇ − 613₇ = 5325₇; 372₈ − 77₈ = 273₈; 1827₉ − 65₉ = 1752₉. (p.48)

2.1 Problem Solving

• Convert percentages to quantities first (e.g. 25% of 200 = 50), convert to the required base, then add or subtract as asked. (p.49)
• Check the answer by converting back to base ten (e.g. 80 + 50 = 130₁₀ = 2002₄). (p.49)