Question

Write the following base 5 numbers in base 10.a. 243_5b. 2000_5

Answer 1

Explanation

Let's imagine that we have a number abcd in base B where a, b, c, d and B are all integers. Then this number in base ten is given by:

$a\cdot B^3+b\cdot B^2+c\cdot B^1+d\cdot B^0$

So as you can see you just need to take each digit and multiply it by a power of B. The exponent of the power depends on the position of the digit. The first digit at the left of the comma has 0 as its exponent, the second has 1, the third has 2 and so on.

Now in part a we must convert 243 in base 5 to its equivalent in base 10. Applying what we just saw we get:

$243_5=2\cdot5^2+4\cdot5^1+3\cdot5^0=2\cdot25+4\cdot5+3\cdot1=50+20+3=73$

So the first number is 73.

In part b the number is 2000 so we get:

$2000_5=2\cdot5^3+0\cdot5^2+0\cdot5^1+0\cdot5^0=2\cdot125+0+0+0=250$

So the second number is 250.

Answer