Base Two to Base Ten: Unsigned Base Two Binary Number 10 1010 1010 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0010 Converted and Written as a Base Ten Natural Number (Positive Integer, Without Sign), in Decimal System

Unsigned base two binary number 10 1010 1010 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0010₍₂₎ converted and written as a base ten number

Conversions:

Convert unsigned binary numbers (base two) to positive integers in the decimal system (base ten)

1. Map the unsigned binary number's digits versus the corresponding powers of 2 that their place value represent.

2. Multiply each bit by its corresponding power of 2 and add all the terms up.

10 1010 1010 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0010₍₂₎ =

(1 × 2⁵⁷ + 0 × 2⁵⁶ + 1 × 2⁵⁵ + 0 × 2⁵⁴ + 1 × 2⁵³ + 0 × 2⁵² + 1 × 2⁵¹ + 0 × 2⁵⁰ + 1 × 2⁴⁹ + 0 × 2⁴⁸ + 1 × 2⁴⁷ + 0 × 2⁴⁶ + 1 × 2⁴⁵ + 0 × 2⁴⁴ + 0 × 2⁴³ + 0 × 2⁴² + 0 × 2⁴¹ + 0 × 2⁴⁰ + 0 × 2³⁹ + 0 × 2³⁸ + 0 × 2³⁷ + 0 × 2³⁶ + 0 × 2³⁵ + 0 × 2³⁴ + 0 × 2³³ + 0 × 2³² + 0 × 2³¹ + 0 × 2³⁰ + 0 × 2²⁹ + 0 × 2²⁸ + 0 × 2²⁷ + 0 × 2²⁶ + 0 × 2²⁵ + 0 × 2²⁴ + 0 × 2²³ + 0 × 2²² + 0 × 2²¹ + 0 × 2²⁰ + 0 × 2¹⁹ + 0 × 2¹⁸ + 0 × 2¹⁷ + 0 × 2¹⁶ + 0 × 2¹⁵ + 0 × 2¹⁴ + 0 × 2¹³ + 0 × 2¹² + 0 × 2¹¹ + 0 × 2¹⁰ + 0 × 2⁹ + 0 × 2⁸ + 0 × 2⁷ + 1 × 2⁶ + 0 × 2⁵ + 0 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 0 × 2⁰)₍₁₀₎ =

(144 115 188 075 855 872 + 0 + 36 028 797 018 963 968 + 0 + 9 007 199 254 740 992 + 0 + 2 251 799 813 685 248 + 0 + 562 949 953 421 312 + 0 + 140 737 488 355 328 + 0 + 35 184 372 088 832 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 64 + 0 + 0 + 0 + 0 + 2 + 0)₍₁₀₎ =

(144 115 188 075 855 872 + 36 028 797 018 963 968 + 9 007 199 254 740 992 + 2 251 799 813 685 248 + 562 949 953 421 312 + 140 737 488 355 328 + 35 184 372 088 832 + 64 + 2)₍₁₀₎ =

192 141 855 977 111 618₍₁₀₎

The number 10 1010 1010 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0010₍₂₎ converted from an unsigned binary (in base 2) and written as a positive integer (with no sign) in decimal system (in base ten):
10 1010 1010 1010 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 0010₍₂₎ = 192 141 855 977 111 618₍₁₀₎

Spaces were used to group digits: for binary, by 4, for decimal, by 3.

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How to convert unsigned binary numbers from binary system to decimal? Simply convert from base two to base ten.

To understand how to convert a number from base two to base ten, the easiest way is to do it through an example - convert the number from base two, 101 0011₍₂₎, to base ten:

Write bellow the binary number in base two, and above each bit that makes up the binary number write the corresponding power of 2 (numeral base) that its place value represents, starting with zero, from the right of the number (rightmost bit), walking to the left of the number, increasing each corresponding power of 2 by exactly one unit each time we move to the left:
powers of 2: 6 5 4 3 2 1 0

digits: 1 0 1 0 0 1 1
Build the representation of the positive number in base 10, by taking each digit of the binary number, multiplying it by the corresponding power of 2 and then adding all the terms up:
101 0011₍₂₎ =
(1 × 2⁶ + 0 × 2⁵ + 1 × 2⁴ + 0 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰)₍₁₀₎ =
(64 + 0 + 16 + 0 + 0 + 2 + 1)₍₁₀₎ =
(64 + 16 + 2 + 1)₍₁₀₎ =
83₍₁₀₎
Binary unsigned number (base 2), 101 0011₍₂₎ = 83₍₁₀₎, unsigned positive integer in base 10