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Q: What is the difference between grace digita wi-fi radio gdi-irc6000w and gdi-irc6000?
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What is special about the multiples of 9?

If I learned why this is true, then I must've forgotten. I did an Internet search on sum of digits and came up with a bunch of depreciation and loan pay-off formulas. So I sat down to try to figure it out.Let's take a 2-digit number N and the digits are ab, where a & b are digits 0-9. The value of N = 10*a + bNot knowing if it's a multiple of nine, set a + b = 9, and let's see what happens.Since a+b=9, then b=9-a, and substitute this into the first equation:N = 10*a + 9 - a = 9*a + 9 --> 9*(a + 1). Since a is a whole number then (a+1) is also a whole number, so N is a multiple of 9. So by setting the sum of the digits equal 9, the number is constrained to be a multiple of nine. I know this isn't a formal proof, but it does show how it works.When you get to 99, the sum is not nine but a multiple of nine. Any higher numbers, and you have 3 digits and more to work with. There should be some sort of general proof out there somewhere for any-digit numbers, but this is a start.--------------I found the following on The Math Forum @ Drexel:Sum of Digits of Multiples of NineDate: 08/12/2004 at 05:19:50 From: SabaSubject: number theory: multiples of 9Why is it that when you add the individual digits of any multiple ofnine until a single digit answer is reached the answer is always nine?Is it possible to prove this?For example, 99 => 9 + 9 = 18 => 1 + 8 = 9Why doesn't it work with other numbers between 1-9 either?Date: 08/12/2004 at 10:10:23From: Doctor LuisSubject: Re: number theory: multiples of 9Hi Saba,Good job finding that pattern! The reason is that the sum of thedigits of ANY multiple of 9 is also a multiple of 9. Since you keepadding the digits (each time getting a new multiple of 9, but asmaller multiple), eventually you'll end up with a single digitnumber. Eventually you'll get to the multiple 9 itself.Now, how do I know that the sum of the digits is always a multiple of9? Suppose that a number N has digits a,b,c,d,...(from right to left),N = a + 10b + 100c + 1000d + ...= a + (b + 9b) + (c + 99c) + (d + 999d) + ...= (a + b + c + d + ...) + (9b + 99c + 999d + ...)= (a + b + c + d + ...) + 9*(b + 11c + 111d + ...)N = (sum of digits of N) + 9 * (some number)Now, look at that equation carefully. It means that(sum of digits of N) = N - 9 * (some number)Since N is assumed to be a multiple of 9, we can write it in terms ofanother integer k, so that N = 9k(sum of digits of N) = 9 * k - 9 * (some number)= 9 * (k - (some number))= 9 * (some other number)Since we showed that the sum of the digits is 9 times some integer,then it is also a multiple of 9 itself.To summarize, starting from a multiple of 9, you keep adding thedigits, each time arriving to a multiple of 9. This establishes achain of decreasing multiples of 9, until eventually you reach 9 (froma two-digit multiple). Does that make sense?It doesn't work for other integers because the chain is broken. Forexample, multiples of 8 such as 56 don't add up to a multiple of 8.


Does everyone see the same color?

The short answer: Probably not, but most people still get the same emotional experience from looking at the same colors. To really address this question in detail, however, we need to consider two factors that affect how we see color. The first factor is the rods and cones in the retina of our eyes—the “hardware” that detects light. Rods detect light and dark, and cones detect color. There are three types of cones, which individually detect red, green, and blue. Together, these cells (called photoreceptors) collect the information we need to distinguish colors.The rods and cones relay information to our brain, which interprets that information. The brain is the second factor—the “software" that gives us the experience of seeing color. Differences in our “hardware" or “software" can give us a different sense of color from what other people experience.The most obvious example of a hardware issue is color blindness (also known as color deficiency), which occurs when one or more cone cells are absent or non-functional. People with color deficiencies might have trouble detecting a certain color. Colorblind men, for instance, usually have a weakness in their green photoreceptors, so they’re less sensitive to differences between shades of green and red.Some people also have additional photoreceptors, and as a result, they’re more sensitive to minor differences in color. These people are called tetrachromats, which literally means “four colors.” Tetrachromats are extremely rare, and scientists assume that only women can be tetrachromats. Men are much more likely to be colorblind, as the genes for red and green cones are carried on the X-chromosome. Women, who have two X-chromosomes, are more likely to be tetrachromats.That covers the hardware issue, but that’s probably not what you’re asking about; you’re wondering whether you see the same “blue" as your friend, provided that you and your friend have the same physical ability to perceive color. We’ve all considered this question in some form, and fortunately, so have scientists."I would say recent experiments lead us down a road to the idea that we don't all see the same colors," color vision scientist Jay Neitz of the University of Washington told LiveScience.One of those experiments involves male squirrel monkeys, but to understand it, we’ve got to learn a little about squirrel monkey biology. Feel free to skip this next paragraph if you’re already a squirrel monkey expert. Squirrel monkeys only have green- and blue-sensitive cones. That means they’re unable to detect the colors green and red against a gray background. Most colorblind humans have trouble accomplishing the same task, since they’ve got the same faulty hardware. Here’s where things get interesting: Scientists injected male squirrel monkeys with a virus that randomly turned some of their cones into red receptors. The monkeys were eventually able to successfully find red and green dots against a grey background—something that they weren’t biologically wired to do. Eventually, physicians might be able to use genetically modified viruses to treat some forms of colorblindness, and the therapy might even be able to give us additional color receptors.However, the experiment also yielded something interesting on the software side: The monkeys' brains found a way to interpret the red photoreceptor cells."The ability to discriminate certain wavelengths arose out of the blue, so to speak—with the simple introduction of a new gene,” explained Joseph Carroll of the Medical College of Wisconsin (note that he was careful to use the phrase “out of the blue”). “Thus, the circuitry there simply takes in whatever information it has and then confers some sort of perception."Carroll believes the experiment shows that color is subjective. Every person develops a unique perspective of color, so what one person experiences as “red" might be what another person experiences as a totally different color. Your brain invented your perception of color—it wasn’t hardwired from birth. With that said, other research indicates that colors prompt similar emotional responses from different people. The color red is better at grabbing our attention than the color blue, and some studies suggest that it can heighten aggression and competitiveness.Blue and yellow light, in particular, tend to affect humans (and other animals) in the same ways—additional photoreceptors, called melanopsin, gauge their levels and send them to areas of the brain involved with emotions. Blue tends to calm us; yellow tends to make us more alert.That’s the long answer, and it’s enough to make your head spin. Yes, we probably see colors differently, but ultimately, it doesn’t really change how we experience them.


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