NAV / 1-Day Return. 20.27 / 0.10% Total Assets. Expense Ratio Adjusted Expense Ratio excludes certain variable investment-related expenses, such as interest from borrowings.

• Motrix 1 1 3 Equals Inches
• Motrix 1 1 3 Equals Kilograms
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B-c = (1–2+3–4+5–6⋯)-(1+2+3+4+5+6⋯) Because math is still awesome, we are going to rearrange the order of some of the numbers in here so we get something that looks familiar, but. Partners: Hope-Hunter Duo Is A ‘One Plus One Equals Three’ HPE GreenLake Game Changer ‘I think the appointment of George Hope working with Paul Hunter is a huge deal,’ says Advizex.

How Can 0.999... = 1?

There are many different proofs of the fact that '0.9999...' does indeed equal 1. So why does this question keep coming up?

Students don't generally argue with '0.3333...' being equal to ¹/₃, but then, one-third is a fraction. Maybe it's just that it 'feels' 'wrong' that something as nice and neat and well-behaved as the number '1' could also be written in such a messy form as '0.9999...' Whatever the reason, many students (me included) have, at one time or another, felt uncomfortable with this equality.

One of the major sticking points seems to be notational, so let me get that out of the way first: When I say '0.9999...', I don't mean '0.9' or '0.99' or '0.9999' or '0.999' followed by some large but finite (limited) number of 9's'. The ellipsis (the 'dot, dot, dot' after the last 9) means 'goes on forever in like manner'. In other words, '0.9999...' never ends. There will always be another '9' to tack onto the end of 0.9999.... So don't object to 0.9999... = 1 on the basis of 'however far you go out, you still won't be equal to 1', because there is no 'however far' to 'go out' to; you can always go further.

'But', some say, 'there will always be a difference between 0.9999... and 1.' Well, sort of. Yes, at any given stop, at any given stage of the expansion, for any given finite number of 9s, there will be a difference between 0.999...9 and 1. That is, if you do the subtraction, 1 – 0.999...9 will not equal zero. But the point of the 'dot, dot, dot' is that there is no end; 0.9999... is inifinte. There is no 'last' digit. So the 'there's always a difference' argument betrays a lack of understanding of the infinite. (That's not a 'criticism', per se; infinity is a messy topic.)

Proof by geometric series

The number '0.9999...' can be 'expanded' as:

0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...

In other words, each term in this endless summation will have a '9' preceded by some number of zeroes. This may also be written as:

That is, this is an infinite geometric series with first term a = ⁹/₁₀ and common ratio r = ¹/₁₀. Since the size of the common ratio r is less than 1, we can use the infinite-sum formula to find the value:

So the formula proves that 0.9999... = 1.

Note: Technically, the above proof requires that some fairly advanced concepts be taken on faith. If you study 'foundations' or mathematical philosophy (way after calculus), you may encounter the requisite theoretical constructs. Copyright © Elizabeth Stapel 2006-2011 All Rights Reserved

Other pre-calculus arguments

Argument from precedence: If you haven't already learned that¹/₃ = 0.333... in decimal form, you can prove this easily by doing the long division:

...and so forth, ad infinitum.

So¹/₃ + ¹/₃ + ¹/₃ = 3( ¹/₃ ) = 1. Reasonably then, 0.333... + 0.333... + 0.333... = 3(0.333...) should also equal 1. But 3(0.333...) = 0.999.... Then 0.999... must equal 1.

Argument from arithmetic: When you subtract a number from itself, the result is zero. For instance, 4 – 4 = 0. So what is the result when you subtract 0.999... from 1? For shorter subtractions, you get:

Then what about 1.000... – 0.999...? You'll get an infinite string of zeroes. 'But,' you ask, 'what about that '1' at the end?' Ah, but 0.999... is an infinite decimal; there is no 'end', and thus there is no '1 at the end'. The zeroes go on forever. And 0.000... = 0.

Then 1 – 0.999... = 0.000... = 0, and 1 = 0.999....

Argument from philosophy: If two numbers are different, then you can fit another number between them, such as their average. But what number could you possibly fit between 0.999... and 1.000...?

Argument from algebra: The expression 0.999... is some number; it has some value. Call this numerical value 'x', so 0.999... = x. Multiply this equation by ten:

x = 0.999...
10x = 9.999...

Subtract the former from the latter:

Solving, we get x = 1, so 0.999... = 1.

'But,' you ask, 'when you multiply by ten, that puts a zero at the end, doesn't it?' For finite expansions, certainly; but 0.999... is infinite. There is no 'end' after which to put that zero.

Argument from semantics: A common objection is that, while 0.999... 'gets arbitrarily close' to 1, it is never actually equal to 1. But what is meant by 'gets arbitrarily close'? It's not like the number is moving at all; it is what it is, and it just sits there, looking at you. It doesn't 'come' or 'go' or 'move' or 'get close' to anything.

On the other hand, the terms of the associated sequence, 0.9, 0.99, 0.999, 0.9999, ..., do 'get arbitrarily close' to 1, in the sense that, for each term in the progression, the difference between that term and 1 gets smaller and smaller. No matter how small you want that difference to be, I can find a term where the difference is even smaller.

This 'getting arbitrarily close' process refers to something called 'limits'. You'll learn about limits later, probably in calculus. And, according to limit theory, 'getting arbitrarily close' means that they're equal: 0.999... does indeed equal 1.

Note regarding all of the above: To a certain extent, each of these arguments depends on a basic foundational doctrine of mathematics called 'The Axiom of Choice'. A discussion of the Axiom of Choice is well beyond anything we could cover here, and is something that most mathematicians simply take on faith.

Top Return to Index

The combination of George Hope as the new Hewlett Packard Enterprise worldwide channel chief and Paul Hunter as the new managing director of North America is an HPE GreenLake channel sales game changer, partners told CRN.

“I think the appointment of George Hope working with Paul Hunter is a huge deal,” said C.R. Howdyshell, president of Advizex Technologies, No. 110 on the CRN 2020 Solution Provider 500, a top HPE GreenLake enterprise partner. “Putting George Hope in that role demonstrates HPE’s significant commitment to the channel. George teaming with (former HPE Worldwide Channel Chief) Paul Hunter amounts to one plus one equals three for HPE’s GreenLake strategy. Those two together are going to drive shoulder to shoulder selling with partners on GreenLake.”

Howdyshell called Hope a 22-year channel veteran who took the top channel job Oct. 1, a “channel execution machine” who knows how to scale a business and Hunter, an 18-year HPE veteran, a proven channel sales advocate who will drive more teaming with partners in the North America sales trenches.

Howdyshell sees GreenLake sales momentum accelerating in HPE’s new fiscal year which begins Nov. 1. Besides the Hope-Hunter combination, HPE partners like Advizex are also benefitting from blockbuster GreenLake deals like the launch of the new customer edition of SAP HANA Enterprise Cloud on HPE GreenLake. “That’s going to be a big play for partners like us who have a strong SAP knowledge set,” he said.

Under Howdyshell’s leadership, Advizex has embraced an everything as a service model, embracing emerging technologies including container based solutions, DevOps and artificial intelligence.

The future for Advizex is providing a full HPE GreenLake everything-as-a-service model around offerings like SAP, VMware, containers, hyperconverged in a consumption based model on premise, said Howdyshell. “This is all about having conversations with customers on their business challenges and how we can help solve those,” he said.

The Hope-Hunter sales charge comes with HPE poised to launch a massive GreenLake sales blitz, tripling its channel investment in the new fiscal year to power big on-premises pay-per-use sales growth, said HPE Senior Vice President and GreenLake Cloud Services General Manager Keith White told CRN recently.

The sales blitz includes a sizable investment in new inside GreenLake sales reps worldwide charged with working hand in hand with partners and an increased account coverage model for partners, both aimed at driving new logo wins.

Paul Cohen, vice president of sales for New York-based PKA Technologies Inc., one of HPE‘s top Platinum partners, said he is also “extremely thrilled and excited” about the channel sales impact of the Hope-Hunter combination.

“The two of them together have dozens of years of experience building efficient and profitable channel programs that work for the mutual benefit of partners, HPE and more importantly customers,” Cohen said. “HPE direct sales and partner alignment have been good, but it has the potential to be something really great in 2021 with Paul Hunter and George Hope. We’re very much looking forward to increased collaboration with the HPE field sales teams.”

Cohen, who was a senior HPE sales executive for nearly seven years before joining PKA, said he is looking forward to meeting with Hope and Hunter to share ideas on “how to drive GreenLake sales growth for the betterment of both HPE and the channel.”

Rob Schaeffer, president of Orange, Calif-based CBT, an HPE Platinum partner that has been recognized for its groundbreaking Refinery of the Future solution, said he also sees Hope-Hunter having a big channel sales impact in 2021.

Motrix 1 1 3 Equals Inches

“Have we in the channel ever had a one two punch like Paul and George both coming from channel backgrounds?” asked Schaeffer. “The answer is no. Do we think that is going to be good for us in the channel? Absolutely. They are dear friends and they are dear friends because they have invested in us and with us. They know our businesses. They know our business model. They don’t have to be sold on the value of the channel. It is no longer a channel tax. It is a channel value-add. Those two are uniquely qualified. They both have a deep understanding of what the HPE ecosystem does well when we do it together.”

Schaeffer said he sees the global pandemic and work from home accelerating high value solutions like those provided by CBT for refinery of future with IT (Information Technology)/OT (operational technology) convergence.

Motrix 1 1 3 Equals Kilograms

“We believe that in 2021 there is going to be a separation of the volume players and the value players,” he said. “The infrastructure business is always going to require speed, efficiency and price from pick, pack and ship big box movers. What we see is a bigger requirement for us in the channel to have areas of specialization that will hit the mark for the customer community. That will be driven by vertical, geography and business outcomes.”

Motrix 1 1 3 Equals How Many Tablespoons

CBT is “extremely bullish” about the potential for 2021 given the new fiscal year go to market planning of Hunter and Hope, said Schaeffer. “The partners are going to be really accretive to the things that HPE is doing,” he said. “If we align and execute correctly this could be not just a one plus one equals two or three, but a one plus one equals five or ten.”