LESSON
02 – CONVERTING TO BASE 10
LESSON NOTE
CONVERTING FROM ANY BASE TO DECIMAL
OPTION 1 – SCIENTIFIC NOTATION
Do you remember how to write a number in
scientific notation?
Let’s take the number 4327 and express
it using scientific notation.
4327
= (4 x 1000) + (3 x 100) + (2 x 10) + (7 x 1)
which is the same as
4327
= (4 x 103) + (3 x 102) + (2 x 101) + (7 x
100)
Let’s consider the value of each number.
The 7 represents the1s. The 2
represents the 10s. The 3 represents the 100s. The 4 represents the 1000s.
It turns out this this will be useful to
convert a number from any base to base 10.
USING SCIENTIFIC NOTATION WITH OTHER BASES
Let’s consider the number 4253 in base
7. We can convert this number by
expressing it in its scientific notation.
Before we can do that, we need to
understand the value of each digit for any number in any base. The rightmost digits always represents the
units (the value 1). The next digit on
the left represents a value that is equal to the base we are working in. The next is the base squared. The one after is the base cubed. And so on…
So, for base 7, the values of the digits
are all powers of 7.
- The rightmost digit represents the 1s.
- The next digit on the left represents the 7s.
- The next digit on the left represents the 49s.
- The next digits on the left represents the 343s.
But for base 4, the values of the digits
are all powers of 4.
- The rightmost digit represents the 1s.
- The next digit on the left represents the 4s.
- The next digit on the left represents the 16s.
- The next digits on the left represents the 64s.
Now back to our number 42537.
We can express it in scientific
notation:
42537 = (4 x 343) + (2 x 49)
+ (5 x 7) + (3 x 1)
Which can also be written as:
42537 = (4 x 73) +
(2 x 72) + (5 x 71) + (3 x 70)
Example 1
Convert the binary (base 2) number to
decimal:
01101001
Solution
01101001 =
(1 x 26) + (1 x 25) + (1 x 23) + (1 x 20)
=
64 + 32 + 8 + 1
Example 2
Convert the octal (base 8) number to
decimal:
726
Solution
726 =
(7 x 82) + (2 x 81) + (6 x 80)
=
(7 x 64) + (2 x 8) + (6 x 1)
=
448 + 16 + 6
=
470
The approach used in the example above
can be summarized in this way.
- First you need to know the base of the number
that you wish to convert to decimal. For binary, the base is
2.
- You multiply the rightmost number by base
exponent zero.
- Add the multiplication of the next number (on
the left) and base exponent one.
- Add the multiplication of the next number (on
the left) and base exponent two.
- Add the multiplication of the next number (on
the left) and base exponent three.
- Keep on going until there is no next number.
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