# Convert the following decimal to both binary and hexadecimal. How do you interpret $-27_{10}$?

I'm having trouble with this problem because I don't know how to interpret the question. I'm given three numbers:

$-27_{10}$, $-128_{10}$, and $150_{10}$.

How do I interpret these numbers? What does the subscript of 10 mean, and how does it make these values decimals?

Thank you.

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It just means that they are written in base 10. Often used when dealing with bases, so that you know: ${123}_7 = 1\times 7^2 + 2\times 7 + 3$ – Thomas Andrews Jan 18 '13 at 21:24
In connection with @Thomas’s comment, note that the subscripted base is itself to be interpreted in base ten. To avoid any possible ambiguity on that score, some authors write it in words when the base is small, e.g., $101_{\text{two}}$ or $10_{\text{five}}$. – Brian M. Scott Jan 18 '13 at 21:35

The subscript is used to inform you that these are numbers in "base 10". $-27_{10}$ is $-27$ as we are accustomed: $-2\cdot 10 + -7\cdot 1$. Subscripts allow you to make clear the base in which the number is represented.

E.g., If we saw only $1001$ and the context in which it appears doesn't make it apparent what the base is taken to be, we wouldn't know if that was $$1001_{10} = 10^3 + 1\times 10^0$$ or $$1001_2 = 1\times 2^3 + 0\times 2^2 + 0\times 2^1 + 1 \times 2^0 =1\times 8 + 0 \times 4 + 0 \times 2 + 1 \times 1 = 9_{10}$$ or $$1001_{16} = 1\times 16^3 + 1 \times 16^0 = 49_{10}$$

For some resources to help with understanding how to convert to and from base $10$ to bases $2$ and $16$, you might want to view this youtube video tutorial (Khan Academy), and this Practical Guide to Decimal, Binary and Hexidecimal Change in Bases.

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Also see Wikipedia: Hexadecimal and Binary Numeral System for helpful discussions of these number systems, and tips for conversion. – amWhy Jan 19 '13 at 15:00

$$-27_{10}=-(2\cdot10^1+7\cdot10^0)=-(1\cdot2^4+1\cdot2^3+1\cdot2^2+1\cdot2^0)=-11011_2$$ similarly, $-128_{10}=-10000000_2$, and $150_{10}=1001110_2$ for writing a number in base 2 we use only two ciphers 0 and 1. In hexadecimal system base is 16 and for writing a number in that system we use ciphers $0,1,2,3,4,5,6,7,8,9,A=10,B=11,C=12,D=13,E=14,F=15$ $$-27_{10}=-(1\cdot16^1+11\cdot16^0)=-1B_{16}$$

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I fixed the zero in the top line, but $150_{10}$ can't end in $1$ in base 2, as it is even. $150_{10}=1001110_2$ – Ross Millikan Jan 19 '13 at 19:15
Now looks great, Adi. +1 – Babak S. Jan 21 '13 at 10:33

The subscript is the numeric base they're written in. For example 5 in binary is 101, then $5_{10}=101_2$

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