⌚ August 25, 2014
Because binary is the language of computers and digital devices, such as watches and calculators, it is beneficial to understand. As an example, how does a specific pattern of zeros and ones, such as 11000000, represent the decimal number 192 in the real world?
To understand this concept in binary, let us first examine how we express values in the decimal number system that we are familiar with.
Decimal Place Value
Suppose we have the decimal number 192. Each digit represents a value based upon its position in the entire number. Digits further to the left have more “weight” or value that the digits to the right. This is called a place value system because the value of each digit is determined by its location.
[ SIDENOTE: “And” is the decimal point ]
We read the number 192 as “one hundred, ninety-two,” not “one hundred AND ninety-two.” Using the conjunction “and” when reading integers (all positive and negative non-decimal numbers plus zero) is incorrect because, in mathematics, the word “and” represents the decimal point.
The phrase “one hundred and ninety-two” is short for saying “one hundred and ninety-two hundredths,” which translates to 100.92. This is different from 192, so pay attention to correct usage.
[ End Sidenote ]
We can see the values of each place by forming a place value chart. Beginning with 1, add a new place to the left. To find the value of the new place, multiply the place immediately to its right by the base of the number system.
For base 10, begin with 1. This is called the ones or units place. Add a new place to its left. To find its value, multiply 1 (the place immediately to its right) by 10 (for a base 10 system). 1 x 10 equals 10, so the new place is called the tens place.
For the third place, multiply 10 (the previous place) by 10 (base 10) to form the hundreds place.
The fourth place is the thousands place (100 x 10 = 1,000), the fifth place is the ten-thousands place (1,000 x 10 = 10,000), the sixth place is the hundred-thousands place (10,000 x 10 = 100,000), and this may continue infinitely.
Creating a decimal place value chart by multiplying the previous place by 10.
Another way to create the place value chart is to begin with the base raised to the 0 power. For base 10, this is 10^0. Then, for each successive place, keep the base the same, but increment the power by 1.
Same decimal place value chart created using 10 raised to successive powers. With base 10, the power indicates how many zeros to append after the 1.
Right-align the number 192 with the table so that the 2 is located in the ones place, the nine is located in the tens place, and the 1 is located in the hundreds place.
The number 192 right-aligned in a decimal place value chart.
The value of the number 192 is the sum of the products of each digit’s place in the table. The 1 means one hundred (1 x 100). The 9 means nine tens (9 x 10), and the 2 means two ones (2 x 1). We sum the product of each place to find the value of the numeric expression 192.
1 x 100 = 100 9 x 10 = 90 2 x 1 = 2 ------------------ Value = 192
Showing why 192 has a value of 192 in decimal.
Of course, this seems obvious in decimal because we are already familiar with base 10. The importance of this technique becomes apparent when we use it to convert between number systems.
Binary Place Value
We can create a binary place value chart using the same technique. Even though the base is 2, the principle remains the same. Let’s create a base 2 place value chart that extends to four bits. We will use the power method for simplicity.
Binary place value chart expanded to four places.
We can expand the binary place value chart to any number of places as needed. Each place to the left is double the previous place. Here, we see a binary place value chart expanded to eight places, which is common when representing one byte.
Binary place value chart for eight places.
Just like the decimal system, we use digits to occupy places according to the place value chart. However, in binary, we are limited to 0 and 1, so we can only use 0 or 1 with each place.
The binary number 1001 right-aligned in a binary place value chart. Always align the number with its rightmost digit located in the rightmost place.
With binary, we cannot use any digits greater than 1, such as 2, 3, 4, 5, 6, 7, 8, 9, A, B, or C. (Yes, letters are used as digits in other number systems that exceed ten numerals.) For example, to represent the decimal number 192 in binary, we cannot use 192 as it is.
We cannot use a decimal number in a binary place value chart and expect it to represent the same value. Conversion is required.
Base 10 and base 2 are two different number systems, so we must convert 192 from decimal into binary in order to represent it properly using base 2 digits.
Will binary allow for a number whose value is equal to 192? Yes, it will, but 192 must first be converted into a pattern of zeros and ones. This involves binary conversion, which we will see in the next lesson.
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