# What is the Binary Number System? - Definition & History

Instructor: Sharon Linde
If you have spent any time around computers, you have probably heard of the binary number system. But what is it, exactly, and why is it so essential to modern computing? In this lesson we'll explore this fascinating topic.

## Binary Number System History

For this lesson, let's look at Amy, a math student. Amy's instructor started the first day of class by telling the students that the invention of computers started in 1703, when Gottfried Wilhelm Leibniz published the essay 'Explanation of Binary Arithmetic.' Leibniz wasn't the first to use the concepts of the binary number system, a system that uses just two numbers, 0 and 1, compared to the decimal system that uses ten numbers, 0-9. However, he was the first to use just two Arabic numerals to codify the simplicity and usefulness. Why would he do this, you ask? He did it mainly as an exercise in logic, inspired by the Chinese sovereign Fu Xi, who wrote about binary systems as long as 4,000 years earlier.

That's a lot for a first day of class. Let's check in on Amy and her classmates and see how they're doing.

## Binary Number Progression

The instructor noticed several bored students and decided to shake things up with a difficult question. 'Amy, how would you count to ten in binary?'

Amy was very surprised, but she at least knew the start, 'One.' After that she was stuck. Do you know what comes next? Let's see if you're right.

The teacher continued, 'It might be helpful for you to know that Leibniz showed that the logical progression used in the decimal positional system was exactly the same for the binary system. Namely, when you get to the highest number for a given position, to go one more, you simply change that number to 0 and add 1 to the position on the left. In the decimal system 9 is followed by 10, 19 by 20, and 99 by 100. Since 1 is the highest number in the binary system, it would be followed by 10. Continue from there, please.'

With some hesitation, Amy worked through some more counting in binary, '11, 100, 101, 110, 111, 1000, 1001, 1010, 1011'

The instructor seemed pleased, 'That's 11, so you can stop.'

## Translating Binary to Decimal

The teacher then asked the class, 'If I offered to sell you a \$50 bill for 10101 dollars in the binary system, would that be a good deal?'

Only silence and confused looks answered him, so he chose to put another student on the spot, 'Jameek, would you pay 10101 binary dollars for \$50?'

'It sounds like a big number, so I don't think I would,' answered Jameek.

'Yes, it sounds like a big number, but it's actually not. That's one of the drawbacks of the system.'

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