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Welcome to our Math lesson on Other Number Systems - The Base 2 Number System, this is the third lesson of our suite of math lessons covering the topic of Decimal Number System and Other Numbering Systems, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
As stated earlier, we use a base 10 number system (hence the name decimal), where we use 10 different digits to write all numbers. However, the decimal system is not the only number system used in practice. For example, computers use the base 2 number system to operate. This is done only for convenience, as it is much easier for the machine to handle two types of input signals rather than ten different ones. Hence, computers are built in such a way that they recognize only two types of input voltages: 0 volt (this means zero input or no input) or 5 volt (this means maximum input). The computer then converts the 0 volt input in a number form (the 0 volt input appears on the screen as the digit 0) and the 5 volt input is shown on the screen by the digit 1. Hence, when you see a number on a computer screen like this
It is important to understand that there are 8 consecutive inputs on the computer produced by the power source that are interpreted as
However, this is not the point we want to discuss, as we are here to explain how this base 2 system (also known as binary system) is related to the base 10 system we use our everyday routine. Hence, we can say that basically, the logic of base 2 number system is similar to base 10 number system. The value of digits increases from right to left, just in the decimal system; the only difference is that in the base 2 number system we use only the digits 0 and 1 to write all numbers instead of 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 that we use to write numbers in the decimal system. In this way, a binary number does not have ones, tens, hundreds, thousands etc., but ones, twos, fours, eights, and so on.
In addition, a binary number can be decomposed in powers, but this time in powers of 2 instead of in powers of 10 as occurs in decimal numbers. Thus, if we read the number
we can decompose it as 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20.
We can find which number in the decimal system the above binary number does correspond. Thus, doing the operations, we obtain
What number in the decimal system does the number 1001001 correspond?
We have
We can also convert a decimal number into a binary one by dividing it recurrently by 2 and then taking all remainders of division from the last to the first to write the corresponding binary number. Let's consider an example to clarify this point.
Write the binary form of the base 10 number 157.
We have:
Thus, taking the remainders from the last to the first, we obtain
Proof::
Hence, the solution was correct.
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