Positioning Numeral systems

Содержание и основные понятия

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Alterna · Chapter 14 · page 14.1
Positioning Numeral systems
The base, the alphabet, the developed shape, and the weight of the bits are the basis for the OGE No. 10 and the EGE No.14.
Numeral systems
is the way of recording numbers using a particular set of numbers.
Positioning
The HS value depends on its location: a unit in the category of hundreds of "weights" is 100 times greater than a unit of units.
GEF
PIF
OGE No.10
EGE No. 14
01
What's a Numeral systems?
Definition, alphabet, base
GEF
PIF
Definition
Numeral systems (numeral system)
is a way of recording numbers using symbols (figures) according to certain rules.
numeral system Alphabet
- an orderly set of all the numbers that can be used.
Base of HS (q)
- the number of different numbers in the alphabet.
Position MOP
- the value of the figure depends on its place (seat) in the number.
Analogies
Imagine the house: each apartment has a room, but
value
5 on the 1st floor and 5 on 9th floor, which is quite different (5th entrance vs 95th apartment) as in the Positioning HS: 5 out of 50 and 500 means different "weights" - 5101=50 and 5102=500.
Table of main systems
System
Basis
Figures
Example
Dual
2
0, 1
101₂
Triple
3
0, 1, 2
212₃
Eight.
8
0–7
753₈
Tenth
10
0–9
2024
Sixteenth
16
0–9, A–F
3FA₁₆
02
Weights of discharge and loose shape
Each position has a "weight" — the degree of foundation
Weight
Of which:
aₙ₋₁ aₙ₋₂ … a₁ a₀
the figure ai is in the weight category
qⁱ
:
A = aₙ₋₁·qⁿ⁻¹ + aₙ₋₂·qⁿ⁻² + … + a₁·q¹ + a₀·q⁰
where q is the base of the HS, ai is the figure in the i-routine (account to the right from zero).
Example: 202410
2024₁₀ =
2
·10³ +
0
·10² +
2
·10¹ +
4
·10⁰
= 2·1000 + 0·100 + 2·10 + 4·1
= 2000 + 0 + 20 + 4
Example: 3FA16
3FA₁₆ =
3
·16² +
F
·16¹ +
A
·16⁰
= 3·256 + 15·16 + 10·1
= 768 + 240 + 10 =
1018₁₀
Minimum base
If the number entry contains the figure d, the base of the HS shall be:
strictly greater than d
For example, the number 124 is recorded in the HS with a base of not less than 5 (i.e. the number 4 is the largest).
03
Positioning properties
Disparities, number of digits, residues - key for EEG 14
Sign of division on the basis of HS
Number in the base count system
q
:
Shared by q
The final figure is 0.
Shared by qk
The last k digits are 0.
Balance of division by q
= last digit
Balance of division (q-1)
= sum of figures (mod q-1)
Shared by (q-1)
The sum of the figures is divided by (q-1)
Example: 3FA16
Последняя цифра A₁₆ = 10₁₀. Значит, 3FA₁₆ mod 16 = 10 (не делится на 16).
Сумма цифр: 3 + 15 + 10 = 28. 28 mod 15 = 13. Значит, 3FA₁₆ mod 15 = 13.
Последние 2 цифры: FA₁₆ = 250₁₀. 3FA₁₆ mod 256 = 250.
Example: 10.1102 (Dual)
Последняя цифра = 0 ⟹ число чётное (делится на 2).
Сумма цифр: 1+0+1+1+0 = 3. 3 mod 1 = 0 ⟹ делится на (2−1)=1 (всегда верно).
Последние 2 цифры: 10₂ = 2₁₀ ⟹ остаток от деления на 4 = 2.
Number of digits in number
Number of N in HS with base q takes up
k
Figures where:
k = ⌊log_q(N)⌋ + 1
Equivalent: q.(k-1) ≤ N < q.k.
Example
How many numbers are recorded in the number 255 in the 16-year system?
log₁₆(255) = ln(255)/ln(16) ≈ 5.54/2.77 ≈ 2.0. ⌊2.0⌋ + 1 =
3 figures
(FF₁₆).
For EGE No. 14
The tasks on the properties of the positional record are a frequent subject of the EGE No.14, especially: the indicator of division by (q-1) through the sum of the figures, the number of digits through logarithms, the residues from the division.
🔬
Quantity analyser
Enter the number and base — receive all properties
Number
Base q
Введи число и основание.
04
Abaucus: Visualization of weights
Squeeze the beads, see how the number changes.
🧮
Abacous: Bit and Weight
Press the beads or enter the number - see the bit by bit.
Base q
Number of levels
Number
05
Fold-shaped calculator
Enter the number and base — receive degradation by class
📐
Open Form of Number
Number
Basis
Введи число и нажми «Разложить»
06
Guess the counting system.
10 tasks — determine the minimum base by type of number
🎯
Trainer Guess the numeral system.
Identify the minimum base in which the entry is allowed
101101
0/0
07
Mini test
📝
5 random questions
1 / 5
0 ✓