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Atricle Dump - History of the Computer; How Computers Add - A Logical Approach
Financial advisors often find themselves consulting to successful entrepreneurs about how to continue to grow their assets after the business has been sold or taken over through a carefully planned succession strategy. But developing a small business (defined here as having less than $50 million in annual revenues) is not so simple.After the initial burst of business success and survival in the first three years, many small businesses encounter struggles that can leave them feeling isolated. What can assist a 30-year old consulting firm whose personal presence and paper products face a changing world of electronic presence and high tra
To make an adder we must duplicate with a logic circuit the way we add in binary. To add 1+1 we need 3 inputs, one for each bit, and a carry in - and 2 outputs, one for the result (1 or 0), and a carry out, (1 or 0). In this case the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to make up the adder for 1 bit.
Now we go another step, and forget about gates, because now we have a Logic Block, an ADDER. Our computer is designed by using various combinations of lo
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You probably realise that the 'standard' PC code is in 8 bit bytes taking the hex system a stage further. You may also know that processors, and Windows software that runs on them, have progressed from 8 bits to 16 bits to 32 bits to 64 bits. Basically this means the computer can work on 1,2, 4 or 8 bytes at once. Don't worry if this is all Gobbledegook, you don't need it to understand how computers add!
OK now to the Math - cringe time! It's a little more complicated than last time, but if you think logically, like a computer, realising they are really dumb, you will sail through it!
We take a break here to look at a bit of math you may not have heard of - Boolean Algebra. Once again it's really simple, but it shows you how a computer works, and why it is so pedantic!
Boolean Algebra is named after George Boole, an English Mathematician in the 19th Century. He devised the logic system used in digital computers more than a century before there was a computer to use it!
In Boolean Algebra, instead of + and - etc. we use AND and OR to form our logic steps.
For example:-
x OR y = z means if x or y is present, we get z.
However,
x AND y = z means that both x and y need to be present to get z.
We can also consider an XOR (eXclusive OR).
x XOR y=z means that x or y BUT NOT BOTH must be present to get z.
That's it! That's all the math you need to understand how a computer adds. Told you it was simple!
How do we use this logic in the computer? We make up a little electronic circuit called a Gate with transistors and things, so we can work on our binary numbers stored in a register - just a bit of memory. (And that's the last electronics you'll hear about!). We make an AND gate, an OR gate, and an XOR gate.
When we add in decimal, for example 9+3 we get 2 'units' and carry one to the 10s, giving 10+2=12
Remember the binary bit values in Decimal - 1,2,4,8 etc? We start at 0, then 1 in the first bit position, the 1 bit. If we add 1 + 1 binary we have to end up with 10, which has a 1 bit in the second bit position, and a 0 in the first, giving Decimal 2+0=2. This second bit position is formed by a CARRY from the first bit.
To make an adder we must duplicate with a logic circuit the way we add in binary. To add 1+1 we need 3 inputs, one for each bit, and a carry in - and 2 outputs, one for the result (1 or 0), and a carry out, (1 or 0). In this case the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to make up the adder for 1 bit.
Now we go another step, and forget about gates, because now we have a Logic Block, an ADDER. Our computer is designed by using various combinations of log
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OK now to the Math - cringe time! It's a little more complicated than last time, but if you think logically, like a computer, realising they are really dumb, you will sail through it!
We take a break here to look at a bit of math you may not have heard of - Boolean Algebra. Once again it's really simple, but it shows you how a computer works, and why it is so pedantic!
Boolean Algebra is named after George Boole, an English Mathematician in the 19th Century. He devised the logic system used in digital computers more than a century before there was a computer to use it!
In Boolean Algebra, instead of + and - etc. we use AND and OR to form our logic steps.
For example:-
x OR y = z means if x or y is present, we get z.
However,
x AND y = z means that both x and y need to be present to get z.
We can also consider an XOR (eXclusive OR).
x XOR y=z means that x or y BUT NOT BOTH must be present to get z.
That's it! That's all the math you need to understand how a computer adds. Told you it was simple!
How do we use this logic in the computer? We make up a little electronic circuit called a Gate with transistors and things, so we can work on our binary numbers stored in a register - just a bit of memory. (And that's the last electronics you'll hear about!). We make an AND gate, an OR gate, and an XOR gate.
When we add in decimal, for example 9+3 we get 2 'units' and carry one to the 10s, giving 10+2=12
Remember the binary bit values in Decimal - 1,2,4,8 etc? We start at 0, then 1 in the first bit position, the 1 bit. If we add 1 + 1 binary we have to end up with 10, which has a 1 bit in the second bit position, and a 0 in the first, giving Decimal 2+0=2. This second bit position is formed by a CARRY from the first bit.
To make an adder we must duplicate with a logic circuit the way we add in binary. To add 1+1 we need 3 inputs, one for each bit, and a carry in - and 2 outputs, one for the result (1 or 0), and a carry out, (1 or 0). In this case the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to make up the adder for 1 bit.
Now we go another step, and forget about gates, because now we have a Logic Block, an ADDER. Our computer is designed by using various combinations of lo
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In Boolean Algebra, instead of + and - etc. we use AND and OR to form our logic steps.
For example:-
x OR y = z means if x or y is present, we get z.
However,
x AND y = z means that both x and y need to be present to get z.
We can also consider an XOR (eXclusive OR).
x XOR y=z means that x or y BUT NOT BOTH must be present to get z.
That's it! That's all the math you need to understand how a computer adds. Told you it was simple!
How do we use this logic in the computer? We make up a little electronic circuit called a Gate with transistors and things, so we can work on our binary numbers stored in a register - just a bit of memory. (And that's the last electronics you'll hear about!). We make an AND gate, an OR gate, and an XOR gate.
When we add in decimal, for example 9+3 we get 2 'units' and carry one to the 10s, giving 10+2=12
Remember the binary bit values in Decimal - 1,2,4,8 etc? We start at 0, then 1 in the first bit position, the 1 bit. If we add 1 + 1 binary we have to end up with 10, which has a 1 bit in the second bit position, and a 0 in the first, giving Decimal 2+0=2. This second bit position is formed by a CARRY from the first bit.
To make an adder we must duplicate with a logic circuit the way we add in binary. To add 1+1 we need 3 inputs, one for each bit, and a carry in - and 2 outputs, one for the result (1 or 0), and a carry out, (1 or 0). In this case the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to make up the adder for 1 bit.
Now we go another step, and forget about gates, because now we have a Logic Block, an ADDER. Our computer is designed by using various combinations of lo
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How do we use this logic in the computer? We make up a little electronic circuit called a Gate with transistors and things, so we can work on our binary numbers stored in a register - just a bit of memory. (And that's the last electronics you'll hear about!). We make an AND gate, an OR gate, and an XOR gate.
When we add in decimal, for example 9+3 we get 2 'units' and carry one to the 10s, giving 10+2=12
Remember the binary bit values in Decimal - 1,2,4,8 etc? We start at 0, then 1 in the first bit position, the 1 bit. If we add 1 + 1 binary we have to end up with 10, which has a 1 bit in the second bit position, and a 0 in the first, giving Decimal 2+0=2. This second bit position is formed by a CARRY from the first bit.
To make an adder we must duplicate with a logic circuit the way we add in binary. To add 1+1 we need 3 inputs, one for each bit, and a carry in - and 2 outputs, one for the result (1 or 0), and a carry out, (1 or 0). In this case the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to make up the adder for 1 bit.
Now we go another step, and forget about gates, because now we have a Logic Block, an ADDER. Our computer is designed by using various combinations of lo
I've had many active and enthusiastic business team members that were their own worst enemies because they exhibited the classic "flea on a griddle" behavior pattern and jumped around chasing one business opportunity today, and then another one tomorrow without ever putting in enough sustained and focused effort to reasonably give themselves a chance to succeed at any of them.I can really relate to this situation since I briefly fell prey to this same "dog in a meat market" syndrome when I first started my own home based business a few years ago. I caught myself trying to chase several different opportunities at once and
To make an adder we must duplicate with a logic circuit the way we add in binary. To add 1+1 we need 3 inputs, one for each bit, and a carry in - and 2 outputs, one for the result (1 or 0), and a carry out, (1 or 0). In this case the carry input is not used. We use 2 XOR gates, 2 AND gates and an OR gate to make up the adder for 1 bit.
Now we go another step, and forget about gates, because now we have a Logic Block, an ADDER. Our computer is designed by using various combinations of logic blocks. As well as the adder we might have a multiplier (a series of adders) and other components.
Our ADDER block takes one bit (0 or 1) from each number to be added, plus the Carry bit (0 or 1) and produces an output of 0 or 1, and a carry of 0 or 1. A table of the input A, B and carry, and output O and Carry, looks like this:-
With no Carry in:
A B c O C
0 0 0 0 0
1 0 0 1 0
0 1 0 1 0
1 1 0 0 1
With Carry in:
A B c O C
0 0 1 1 0
1 0 1 0 1
0 1 1 0 1
1 1 1 1 1
This is known as a Truth Table, it shows output state for any given input state.
Let's add 2+3 decimal. That is 010 plus 011 binary. We will need 3 ADDER blocks for decimal bit values of 1, 2 and 4)
The first ADDER takes the Least Significant Bit (decimal bit value 1) from each number. Input A will be 0
Input B will be 1
With no carry - 0.
From the truth table this gives an output of 1 and a carry of 0 (3rd row).
BIT 1 RESULT = 1
At the same time the next ADDER (decimal bit value 2) has inputs of A - 1, B - 1 and a carry of 0, giving an output of 0 with a carry bit of 1 (4th row).
BIT 2 RESULT = 0
At the same time the next ADDER (decimal bit value 4) has inputs ofA - 0, B -0 and a carry of 1, giving an output of 1 with no carry - 0 (5th row).
BIT 4 RESULT = 1.
So we have bits 4,2,1 as 101 Binary or 4+0+1=5 decimal.
It seems like a laborious way to do it, but our computer can have 64 adders or more, adding simultaneously two large numbers billions of times a second. This is where the computer scores.
Next time we will get to how a computer performs more complcated operations, and it's simple!
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