Base Conversions

Number Theory and Algebra

• Base Conversions
  • Decimal integer to any base
  • Integer without exponent in base b to decimal
  • Integer with exponent in base b to decimal
  • Fractional decimal number to base b
  • Fractional number in base b to decimal

Source: HP-45 Applications Book (HP 00045-90320 Rev. B Reorder 00045-66001, Dec 1974)

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Base Conversions

Note:
Base conversation algorithms are given for positive values only. To convert a negative number, change sign, convert, and change sign result.

Decimal integer to integer in any base

I₁₀ → J_b

In the following key sequence, f+1 is the number of digits in J_b.
d_i (i=1, …, f+1) represents the i^th digit in J_b, counting from left to right, i.e.

J_b = (d₁d₂ --- d_f+1)_b

For large numbers, J_b = (d₁·d₂ --- d_f+1)_b * b^f, see example 3.

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	b	ENTER ENTER
2	I	STO A LN
3		x<>y LN ÷	D
4		INTG 6'IP	f	Let f be the largest
				integer ⩽ D
5		x<>y ENTER ENTER RCL A
6		R↓ R↓ x<>y y˟ ÷	E1	di = integer part of Ei
				(i=1, …, f)
7	d1	- ×	E2
8	di	- ×	Ei+1	Perform 8 for i=2, …, f
9		DISPLAY 4'ALL	Ef+1

Example 1: Convert 1206 to hexadecimal (base 16).
(The hexadecimal digits are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)

Answer: 1206₁₀ = 4B6₁₆ (f = 2)

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		BASE 2'HEX		hexadecimal mode
2	1206	1 2 0 6 ENTER	4B6h

Example 2: Convert 513 to octal (base 8).

Answer: 513₁₀ = 1001₈

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		BASE 3'OCT		octal mode
2	513	1 2 0 6 ENTER	1001o

Example 3: Convert 6.023 × 10²³ to octal.

Answer: 6.023 × 10²³ = 1.7743₈ × 8²⁶

Note: If we consider 6.023 × 10²³ to be a scientific mesurement good only to 4 significant digits, it is meaningless for the octal representation to contain more than 5 significant digits. Therefore, we stop before the loop is completed.

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	8	ENTER ENTER 6 . 0
2		2 3 E 2 3
3		STO A LN
4		x<>y LN ÷	26.33
5		INTG 6'IP	26	f=26 (Note: this gives
				the exponent in base 8)
6		x<>y ENTER ENTER RCL A
7		R↓ R↓ x<>y y˟ ÷	1.99	d1=1
8	1	- ×	7.94	d2=7
9	7	- ×	7.54	d3=7
10	7	- ×	4.34	d4=4
11	4	- ×	2.69	d5=5 (rounded), stop

Integer without exponent in base b to decimal

(d₁d₂ --- d_n-1d_n)_b → I₁₀

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1	b	ENTER ENTER ENTER
2	d1	×
3	di	+ ×		Perform 3 for i=2, …, n-1
4	dn	×

Examples:

1. 730020461₈ = 123740465₁₀
2. 7D0F₁₆ = 32015₁₀ (A = SIN, B = COS, C = TAN, D = √x, E = y˟, F = 1/x input in hexadecimal mode)

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		BASE 3'OCT		octal input
2	730020461o	7 3 0 0 2
		0 4 6 1
		BASE 7'o	730020461o
3		BASE 1'DEC		decimal mode

LINE	DATA	OPERATIONS	DISPLAY	REMARKS
1		BASE 2'HEX		hexadecimal input
2	7D0Fh	7 SIN 0 1/x
		BASE 6'h	7D0Fh
3		BASE 1'DEC	32015	decimal mode