🎯 Various numeral systems in Scalar Calculator

Today you will learn how to use different numeral systems in the Scalar calculator. Thanks to Scalar it is very easy to:

  • Convert decimal system to other numeral systems
  • Convert other numeral systems into a decimal system
  • Enter values simultaneously in different numeral systems
  • Acquire information about the digits of a given number on the basis of the indicated numeral system

Stay tuned, this is exciting πŸ™‚

⭐ Conversion to other numeral systems

Conversion of the number to different numeral systems is done at the end, i.e. after determining the value of the expression. User indicates in which numeral system the result should be presented. General syntax:

<NumeralSystem> expression

  • binary: bin expression
  • octal: oct expression
  • decimal: expression
  • hexadecimal: hex expression
  • base-n: base.n expression

Please follow the below examples

Scalar - Numeral Systems - Decimal to other base

Scalar code result:

scalar > bin 127
e1 = b.1111111    eps = 0
scalar > oct 127
e2 = o.177    eps = 0
scalar > hex 127
e3 = h.7F    eps = 0
scalar > base.3 127
e4 = b3.11201    eps = 0
scalar > base.16 127
e5 = h.7F    eps = 0
scalar > base.9 (1+3)*4
e6 = b9.17    eps = 0

Scalar script:

bin 127
oct 127
hex 127
base.3 127
base.16 127
base.9 (1+3)*4

⭐ Convert to decimal system

The mathematical engine of Scalar always works in one numeral system, so before it is possible to determine the result of an expression, all values given in different numeral systems are converted to one decimal standard. In Scalar you can enter numbers using any numeral system. General syntax:

<NumericalSystemPrefix.>NumberLiteral

  • binary: b.XXXX
  • octal: o.XXXX
  • decimal: XXXX
  • hexadecimal: h.XXXX
  • base-n: b<n>.XXXX

Please follow the below examples

Scalar - Numeral Systems - Other base to decimal

Scalar code result:

scalar > hex 123
e1 = h.7B    eps = 0
scalar > h.7b
e2 = 123
scalar > 123-h.7b
e3 = 0
scalar > hex h.7b
e4 = h.7B    eps = 0
scalar > bin h.7b
e5 = b.1111011    eps = 0
scalar > b1.11111
e6 = 5
scalar > b1.11111 - 5
e7 = 0

Scalar script:

hex 123
h.7b
123-h.7b
hex h.7b
bin h.7b
b1.11111
b1.11111 - 5

⭐ Mixing different numeral systems in a single expression

Scalar is very flexible, thus there is no limitation in terms of entering number in various numeral systems.

Scalar - Numeral Systems - Mixing bases

Scalar code result:

scalar > b1.11111*5
e1 = 25
scalar > x = b16.7b + b1.11111
scalar > x
e2 = 128
scalar > 2*x
e3 = 256
scalar > hex 2*x
e4 = h.100    eps = 0
scalar > h.100
e5 = 256
scalar > base.1 32
e6 = b1.11111111111111111111111111111111    eps = 0

Scalar script:

b1.11111*5
x = b16.7b + b1.11111
x
2*x
hex 2*x
h.100
base.1 32

⭐ Numerical systems with higher base

Using a standard alphabet, i.e. alphabetical order of letters, you can enter numbers in a numeral system with base between 1 and 36. In case of any larger bases, please use the function base (n, …) that accepts many parameters.

Scalar - Numeral Systems - Any base n

Scalar code result:

scalar > base.3 56
e1 = b3.2002    eps = 0
scalar > base(3, 2, 0, 0, 2)
e2 = 56
scalar > base(256, 2, 0, 0, 2)
e3 = 33554434
scalar > base(256, 54, 125, 36, 2)
e4 = 914170882
scalar > base.23 10000000
e5 = b23.1CGKDE    eps = 0
scalar > b23.1cgkde
e6 = 10000000

Scalar script:

base.3 56
base(3, 2, 0, 0, 2)
base(256, 2, 0, 0, 2)
base(256, 54, 125, 36, 2)
base.23 10000000
b23.1cgkde

⭐ Non integers conversion

During the conversion of the result to the indicated numeral system, only the integer part is converted. Please follow the epsilon remark to be sure what part was not converted.

Scalar - Numeral Systems - Non integers

Scalar code result:

scalar > pi^e
e1 = 22.45915771836104
scalar > bin pi^e
e2 = b.10110    eps = 0.45915771836104113
scalar > e^pi
e3 = 23.140692632779263
scalar > bin e^pi
e4 = b.10111    eps = 0.14069263277926325
scalar > bin -e^pi
e5 = -b.10111    eps = 0.14069263277926325

Scalar script:

pi^e
bin pi^e
e^pi
bin e^pi
bin -e^pi

⭐ Acquiring information about digits

Scalar provides several clever functions that allow to obtain information about the digits of a given number. Long press the “example /?” button and enter “numeral” in the search field. You will be presented with the list of functions.

  • ndig10(x) – number of digits of a number in base 10
  • ndig(x, base) – number of digits of a number in any base
  • dig10(x, position) – digit of a number in base 10 at specified position
  • dig(x, position, base) – digit of a number in any base at specified position
Scalar - Numeral Systems - Working with digits - help

And a few examples:

Scalar - Numeral Systems - Working with digits

Scalar code result:

scalar > base.3 1234567
e1 = b3.2022201111201    eps = 0
scalar > ndig10(1234567)
e2 = 7
scalar > ndig(1234567, 3)
e3 = 13
scalar > dig10(1234567, 5)
e4 = 5
scalar > dig(1234567, 5, 3)
e5 = 2
scalar > dig(h.abfd, 3, 16)
e6 = 15

Scalar script:ο»Ώ

base.3 1234567
ndig10(1234567)
ndig(1234567, 3)
dig10(1234567, 5)
dig(1234567, 5, 3)
dig(h.abfd, 3, 16)

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