Binary for Programmers
Let us run through a basic guide for programmers
I’ve seen many presumptions and assumptions that programmers not only know
Binary, but can easily convert between
Binary and even perform some
Binary mathematical operations.
This Medium post represents the missing guide that we need to help us not only understand
Denary, but be in a position to build upon that knowledge when we implement
Binary operations in our chosen programming language.
Difficulty: Beginner | Easy | Normal | Challenging
- None, but for practical purposes it would be useful to be able to produce a “Hello, World!” application in your chosen programming language
Base number system: A term that means roughly the same as “Number bases”
Binary: The name of the base 2 number system
Decimal: The system for denoting Integer and Non-Integer numbers
Denary: The name of the base 10 numbering system
Hundred: The number equivalent to the product of ten and ten; ten more than ninety; 100
Integer: A number that has no fractional part, that is no digits after the decimal point
Number bases: The number of different digits that a system of counting uses
Radix: An alternative name for number base
Ten: Equivalent to the product of five and two; one more than nine; 10
Unit: An individual thing, the smallest number
Binary Cheat Sheet: The answer
Binary numbers have a pattern, from Denary numbers to Binary numbers. For comparison I’ve made a rather natty table showing
Binary numbers (with
Binary on the right). Here are the first ten in the table:
There are minor complexities here.
Binary only has
1 (as we will see later) so 10 in
Binary is not ten but when we say the
Denary equivalent we would term this to be two.
Whoa, whoa. What does any of this mean?
Ok, let us take this from first principles.
Computers are not human. They run on electricity, that means they have to use electricity to represent information.
It is easier and better to use switches in the computer to store that information. How do we do that?
Why do we count in ten: The mobile example
Here is the iPhone lock screen, and this is how you usually unlock your phone using a passcode
When thinking about this, let us think through a few pertinent questions while thinking of the design.
- How many fingers do you have?
- How many numbers are there on the iPhone screen?
These aren’t difficult questions. There are 10 of each.
We are going to see that 10 is a very important number to the human race, and even has a special name as a numbering system.
This numbering system is our “normal” one for counting and that. We call this
Denary and will explain this in the following section:
Denary part 1: 0–9
Our common numbering system is
Denary. The system has
Units from 0–9 as demonstrated in the following:
We will return to
Denary later to make clear a few principles to do with counting. Let us first make this even easier by explaining
Binary which only allows us to use the numbers 0 and 1. Yes, really — there is a numbering system which only allows allows two numbers.
Let us try to explain this with real life examples.
Binary in Real life: The light switch
Think of a light switch
On a computer the computer saves the information as a single number.
1 = There is power
0 = There is no power
or, to put it another way
We can see all of the options that can be stored on a single switch:
This could be thought as a numbering system that stores two numbers: zero and one.
The light switch numbering system — One light switch
Largest number that can be stored: 1
Smallest number that can be stored: 0
Total number of options that can be stored: 2
But what if we put more than one light switch in a row (or series?)
The light switch numbering system — Two light switches
Largest number that can be stored: 3
Smallest number that can be stored: 0
Toal number of options that can be stored: 4
To demonstrate this, we will go through all of the options for each switch:
Logically we can continue this process. What is the largest number that three switches can show?
I’ll give you the answer after a break about how counting works for
Denary part 2: How to count
When you first start to count you are taught “
Denary” — our “normal” counting system.
A history lesson
Denary begins with the letters Dec.
Decimal, Decade, December and other words beginning with DEC do so as they actually are related to the Latin word decem (meaning ten). In other words (on some level) Dec = Ten!
A counting lesson
The following language is often used with young children when they learn to count.
The Digits 0–9 are called
Units , numbers between 10–90 are
Tens and numbers between 100–900 are
For example 572 has five
Tens and two
Exploring a larger number
Let us take 195, 728, 381. A large number. It contain three
Tens and one
Units. We have names for the larger numbers too —
You can see that you multiply by ten every time a new number is added.
We can see a table of how increasing denary numbers maps to five
We increase the
Units each time in order to count. When the
Units reach 10, we increase the
Tens by one. It then follows that when
Tens reach 10, we increase the
Hundreds by one.
The column on the left shows the total.
Denary part 3: Rounding up
Denary is the counting system we use for most things in our lives. When we go to the shop, we might buy 15 things including 6 bottles of water.
Denary involves the numbers: 0 to 9
Despite the fact only use those numbers we can create much larger numbers out of them, there seems in fact to be no limit!
Some say we use
Denary because we have ten fingers. Others say it is because it is so easy to use. In any case, it seems to have been adopted for most uses in our modern lives.
But what if other numbering systems existed (Hint- they do, and we use them every day).
We are going to explore other numbering systems, that doesn’t use
Denary at all!
Binary: An alternative numbering system
Simple Binary: How to count
There is a numbering system that only uses 0 and 1's.
note that we multiply by two each time a new number is added. If this looks like it is familiar, this is because it is the same for the
Denary equivalent that is shown in a section above.
In order to calculate 1110’s equivalent in
Denary we perform the following calculation: 8 + 4 + 2 + 0
The reason? It is explained by the following diagram, when we add the result of each
Binary place together:
We can show the possible numbers that can be created with four
Binary digits with the following table:
Essentially we are performing addition to create the
Denary equivalent of a
Binary there are four different rules (only!) for addition. Addition (+) works much like in
Denary as we will see:
0 + 0 = 00 + 1 = 1 1 + 0 = 11 + 1 = 10
The only real difficulty for people is the last one.
1 + 1 = 10 makes sense because all of the numbers are their
That is representation of the following
1 + 1 = 10
because the numbers are
Binary representations. Yes, and remember 10 isn’t ten in the world of
Binary — it is two!
So we can make this clear by using a subscript to show the numbers are in
Base two/ Binary
1₂ + 1₂ = 10₂
Binary there are four rules. Now these rules are precisely the rules that apply to
Denary numbers. Since
Binary only has two numbers (zero and one) the solution to each rule shown is the same in
Binary as it is in
0₂ * 0₂ = 0₂0₂ * 1₂ = 0₂1₂ * 0₂ = 0₂1₂ * 1₂ = 1₂
A simple example is shown in the following:
We have covered the basics of addition, multiplication and counting using
Binary numbers. The rules for addition, multiplication are even the same for
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