Don’t go against what you believe the will of Christ is for you.ħ. Don’t go against conscience or usurp God’s sovereignty in your life. People sense the Lordship of Christ in different ways. For whether we live, we live unto the Lord and whether we die, we die unto the Lord: whether we live therefore, or die, we are the Lord’s. For none of us liveth to himself, and no man dieth to himself.Ĩ. He that eateth, eateth to the Lord, for he giveth God thanks and he that eateth not, to the Lord he eateth not, and giveth God thanks.ħ. He that regardeth the day, regardeth it unto the Lord and he that regardeth not the day, to the Lord he doth not regard it. Let every man be fully persuaded in his own mind.Ħ. One man esteemeth one day above another: another esteemeth every day alike. Yea, he shall be holden up: for God is able to make him stand.ĥ. Who art thou that judgest another man’s servant? to his own master he standeth or falleth. Let not him that eateth despise him that eateth not and let not him which eateth not judge him that eateth: for God hath received him.Ĥ. For one believeth that he may eat all things: another, who is weak, eateth herbs.ģ. The Principle of Encroachment: Will it violate the lordship of Christ in my life? Does it violate my conscience? Is this something I believe the Lord would want, or not want, for me? The Principle of Equivocation: Will it be a covering for my sin? Am I doing it in the name of freedom in Christ, or am I pandering to evil desires?ġ Peter 2:16 – As free, and not using your liberty for a cloke of maliciousness, but as the servants of God.Ħ. We must not allow something non-moral to have mastery over us.ĥ. For sin shall not have dominion over you: for ye are not under the law, but under grace. Neither yield ye your members as instruments of unrighteousness unto sin: but yield yourselves unto God, as those that are alive from the dead, and your members as instruments of righteousness unto God.ġ4. Let not sin therefore reign in your mortal body, that ye should obey it in the lusts thereof.ġ3. The Principle of Enslavement: Will it bring me into bondage? Does it have the potential to enslave me? (Again, see Hebrews 12:1 1 Corinthians 9:24-27.) We must exercise self discipline and perseverance.Ĥ. Things that are not wrong in themselves can weigh us down and take away our enthusiasm for the things of God. But I keep under my body, and bring it into subjection: lest that by any means, when I have preached to others, I myself should be a castaway. I therefore so run, not as uncertainly so fight I, not as one that beateth the air:Ģ7. Now they do it to obtain a corruptible crown but we an incorruptible.Ģ6. And every man that striveth for the mastery is temperate in all things. Know ye not that they which run in a race run all, but one receiveth the prize? So run, that ye may obtain.Ģ5. Hebrews 12:1 – Wherefore seeing we also are compassed about with so great a cloud of witnesses, let us lay aside every weight, and the sin which doth so easily beset us, and let us run with patience the race that is set before us,…ġ Corinthians 9:24-27 – 24. The Principle of Excess: Will it slow me down in my spiritual race? The Principle of Edification: Will it build me up or help me mature? (Again, see 1 Corinthians 10:23.)ģ. The Principle of Expedience: Will it be profitable or beneficial to me spiritually?ġ Corinthians 10:23 – All things are lawful for me, but all things are not expedient: all things are lawful for me, but all things edify not.Ģ. Take time to meditate on the principles outlined below.ġ. It’s practical, though MacArthur does use some big words in his attempt to make important points begin with “e.” I passed these tips along to my children and thought I’d share them with you with the hope that you’ll find them encouraging. The sermon is entitled “Making the Hard Decisions Easy.” What follows are my notes from a message about making decisions within the will of God, which was given by John MacArthur on his radio program. It should go without saying that prayer is key for our decision making, too. As for applying those principles, God gave you and me a brain, and He expects us to use it. What the Scriptures do give us is guidance by way of principles. But that doesn’t mean the Scriptures aren’t useful for helping us make decisions and finding God’s will. The Bible isn’t going to tell you which bank to put your money in, whether you should take a certain job, or instruct you on something as trivial as what to have for lunch. But what about those things where the Bible doesn’t give clear cut guidance? Should we make up a set of rules to follow as many Christians do? The Bible speaks clearly about many things which are sinful, such as covetousness, sexual sins, worshipping false gods, and so on.
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No you didn't you idiot you ran for 50 metres and then stopped and you're talking loudly, of course you didn't lose her. You run and to be fair, they ran for like a minute and then kept on stopping asking each other " did we lose her?". One issue I have with it, as soon as they realize that the marksman is actually a single woman, armed with a scoped bolt action rifle, they should have taken her on when they still had the numbers, before they started dying. The film is decent for the first 15 minutes, then it becomes incredibly bad. However from the get go, you just cannot relate to the characters, they are just so dull, uninspiring and you really can't root for them. The film actually starts with some breathtaking shots of the mountains, the woodlands and the nature. This review contains spoilers, click expand to view. It's evident that the script did not give for more and the film tries to fill in gaps with totally useless flashbacks, and ends abruptly with nothing resolved when there's obviously no other place to move forwards. This type of film shamelessly mocks the viewer, because it's more than clear that the creators know in advance what they're doing with their story, and continue to believe when assembling it, that their poor tension built from having their characters running and escaping over and over again This type of film shamelessly mocks the viewer, because it's more than clear that the creators know in advance what they're doing with their story, and continue to believe when assembling it, that their poor tension built from having their characters running and escaping over and over again without sense, knowing that most of them will die, will be enough to justify what they're doing, when in reality the whole thing is just completely boring. I think so far we are doing a good job □ In other words, 99.1% of the variation is explained by our model. This is what we were looking for! Our R-squared is 0.991 or 99.1%. If you go back to our output table, and find the second last row, you will see it says "Multiple R-squared: 0.991". For now, let's assume we have the general case. But, as with everything, it is not always the case, and there are multiple other factors that need to be considered in a detailed regression analysis. Generally, the higher the R-squared, the better. It shows how much of the total variation in the model is explained on a scale of 0% to 100%. R-squared is always between 0% and 100% and determines how close the observations from the dataset are to the fitted regression line.įormula: R-squared = Explained Variation/Total VariationĪfter looking at the formula, you intuitively get it. The statistical measure of the goodness of fit is called R-squared. Goodness of fit of a model basically shows how well it fits the observations from the dataset. So what is the goodness of fit you may ask? In our case both coefficients have a very low p-value (that is close to zero), therefore we can state that our coefficients are significant at 100% confidence level, and we reject the null hypothesis of these variables having no impact on the prediction of the dependent variable. " indicates how many stars represent which level of significance. The row right under the coefficients table "Signif. These starts are created for convenience to indicate the statistical significance. Here are our p-values for the intercept and "beta 1".Īlso notice the stars (***) next to them. I prefer to call the data I work with “mydata”, so here is the command you would use for that: If your you have your own dataset that you would like to practice with by following the steps in this article, you can learn about importing different types of files into R here. Although the step of “loading” this dataset isn’t required, it’s a good practice to get familiar with □ R has a variety datasets already built into it. Interpreting linear regression coefficients in R.Basic analysis of regression results in R.Now it’s time to run some regressions in R!īelow are the steps we are going to take to make sure we do learn how to do linear regression in R: Forecast an outcome using the existing model but given a new independent variable value (new X value).īelow I will show how to do simple linear regression in R using a dataset built into R as well as provide basic regression analysis.“Strength” of impact of the independent variables (Xs) on dependent variable (Y). Two major insights we are (generally) trying to get from a regression: In the formula above: Y = dependent variable: X = independent variable Beta = coefficient on X Epsilon = error term. If you have more than one independent variable (Xs), there will be more components with beta_ and x_ in the formula, and it is something we call a multiple linear regression. So, paying more for a Red Pitbull is not justified, especially when you know that this type of Pitbull is fairly common. In reality, the red Pitbull is exactly the same as the blue or black Pitbull a Pitbull remains a Pitbull. However, some people mistakenly believe that the Red Pitbull is rarer than others or that it is part of a particular breed thus, they are willing to pay more for this type of Pitbull. Color: The coat color is influenced by the parents, and can be Black and Tan, Brown, Fawn, White or Blue with a merle or brindle pattern Temperament: The Rottweiler Pitbull mix can be strong-willed and stubborn. The Red Pitbull has nothing to do with the Red Nose Pitbull either since we are only talking about the color of its fur. Coat: A short coat that is can be light and fine like the Pitbull or thick and dense like the Rottweiler. The Boxer is usually a fawn or brindle coloring, but often they have black or white markings. flashy seal (black) brindle puppy- 1400 They Pets and Animals Vancouver 1,400. Boxer Pitbull mixes come in an array of colors. The Red Pitbull is not really red its fur is just a tan-brown color. Rehoming Rhodesian Ridgeback Pitbull pups (Colorado Springs) Oct 15. Red Pitbull Image Credit: Voltgroup, Shutterstock As a result, these dogs are generally more expensive than Pitbulls of other colors.ģ. It is possible that it also has some parts where its hair is white, often on the belly level.īesides, brindle Pitbulls are relatively rare finding them in dog shelters is quite an accomplishment. Meet Biscuit, a Boxer & Pit Bull Terrier Mix Dog for adoption, at PCS Pups in Phenix City, AL on Petfinder. So, a black brindle pitbull will have a light base color with moiré patterns. Most of the time, the brindle Pitbull will have a coat made up of several shades of brown, tending to black and forming stripes similar to those of a tiger but darker. You’re going to have a medium-sized, muscular and athletic dog with a silky coat that’s likely radiates a golden hue in the sunlight. American Pit Bull Terriers come in a much wider variety of colors, so this means there is a greater chance of there being a bigger selection of coat colors in a Boxer Pit Bull mix. A post shared by CA Pit bull Rescue is a brindle Pitbull? The term “brindle Pitbull” refers to a Pitbull whose coat is made up of several colors. With the Boxer and American Pit Bull closely related, their puppies have a predictable appearance unlike other mixes such as Pomsky. Quincy celebrated its centennial in 2007. Free agricultural and geology tours are offered, as well as a produce sale, tractor pull, Farm-to-Market fun run, and many other activities. On the day of the celebration, floats created by local schools and farm equipment parade through town. For a month beforehand, the roads leading to town are decorated with signs showing local crops and products made with them. The park is also home to a large field with a baseball diamond, playground, picnic area, and small skate-park.Įvery 2nd Saturday in September, Quincy celebrates Farmer-Consumer Awareness Day. Quincy has several parks including an aqua park with a waterslide with 2 water fed half tubes, heated pools, and special areas for kids. The Gorge Amphitheatre brings a variety of people through the Quincy Valley. The area offers boating, water skiing, hunting, fishing, snow skiing, and concerts. The City of Quincy is located in Central Washington with close proximity to winter and summer activities. The gender makeup of the city was 50.7% male and 49.3% female. 36.6% of residents were under the age of 18 11.5% were between the ages of 18 and 24 26.9% were from 25 to 44 16.6% were from 45 to 64 and 8.3% were 65 years of age or older. The median age in the city was 26.2 years. The average household size was 3.51 and the average family size was 3.87. 15.1% of all households were made up of individuals, and 6.7% had someone living alone who was 65 years of age or older. There were 1,915 households, of which 55.2% had children under the age of 18 living with them, 55.2% were married couples living together, 16.2% had a female householder with no husband present, 9.0% had a male householder with no wife present, and 19.5% were non-families. Hispanic or Latino of any race were 74.3% of the population. The racial makeup of the city was 54.9% White, 0.4% African American, 0.9% Native American, 0.7% Asian, 0.2% Pacific Islander, 40.6% from other races, and 2.4% from two or more races. There were 2,020 housing units at an average density of 407.3 per square mile (157.3/km 2). The population density was 1,360.9 inhabitants per square mile (525.4/km 2). 2010 censusĪs of the census of 2010, there were 6,750 people, 1,915 households, and 1,541 families residing in the city. In the event of a complaint from a sale where finance was the payment method we will work to resolve this in-house with you and in line with the Financial Conduct Authority () & Financial Ombudsman Service () requirements.Fishing, hiking and bird-watching occur at nearby Quincy Lake, a remnant of the temporary Pleistocene lakes that were created by flood waters from glacial Lake Missoula. In the unlikely event that a problem cannot be resolved in-house, you can request arbitration, in writing, from the Secretary, MIRAD, PO Box 333, Southport, PR9 7GW. New Reg is a recognised reseller of DVLA registrations and is registered with the DVLA to supply physical number plates.ĬOMPLAINTS PROCEDURE - Our firm is a Member of the Institute of Registration Agents and Dealers and is governed by their strict rules and regulations. New Reg is not affiliated to the DVLA or DVLA Personalised Registrations. All number plates are subject to availability, E&OE, DVLA transfer fees and our terms and conditions, some number plates are also subject to VAT.ĭVLA is a registered trademark of the Driver & Vehicle Licensing Agency. New Reg sell DVLA registrations (Unissued Government Stock), private registrations owned by third parties and our own stock on a 'first come, first served basis'. Our car registrations 24 hour 'Buy' form takes priority over our 'Enquire' form or a 'Telephone' enquiry. New Reg will coordinate the entire transfer of your personalised registration with DVLA when requested by you. Jay’s strange sense of humor appears regularly, like when he builds a perfect woman who “had an ass like Rosa Acosta” and “smelled like strawberries” on “Rough Love,” or when he’s absorbed by the flaws of Western civilization on “Run and Hide.” The rumors have been justified. Jay’s lines are clever and self-reflective, and his references are evergreen: “Fuck Bill O’Reilly and Rudy Giuliani,” he passionately raps on “New Illuminati.” It’s rewarding to be swept up in his aura, and to feel the magnitude of every strategically placed interlude, every space where the beat rides endlessly, and every roughly mixed verse. Somehow the record doesn’t suffer from the delay, if anything the often drumless production-heavy on somber piano melodies and lush samples-is timeless. When a slightly unfinished version of Jay Electronica’s Act II appeared this October, it had been a little over a decade after its initial slated release, and most fans had given up hope on it ever actually coming out. It was an album that was rumored to be rap’s next opus before it even materialized. Jay Electronica: Act II: The Patents of Nobility (The Turn) The old world that Dogleg wrote about sucks in its own way, but it’s the world they deserve. You air-drum the little hitch in “Fox” again and again across your steering wheel you throw your chest forward in your home-office at all the perfectly executed half-time breakdowns you do isometric lunges while Stoitsiadis sings about disintegrating. We’re left to sit alone and imagine what these songs should be doing. The fact that Melee conjures a year so different from the one we got is part of why it leaves such a mark: The histrionics of emo aren’t just dramatic, they’re now science fiction. Singer-songwriter Alex Stoitsiadis was supposed to be hollering his hooks over melodic post-hardcore guitars in roiling 250-cap clubs, and the scenes of heartbreak he described were supposed to be playing out for listeners in real life. –Simon ReynoldsĪ band’s impact shouldn’t be hypothetical, but here’s Dogleg, the debutants of Michigan emo, whose breakout year mostly took place in the imagination. Twice as Tall triumphs not so much for its substance but as a shimmering surface, a landslide victory for the politics of pleasure. “I no be politician/Me no like no politics,” Burna insists-and that rings true. But these Fela-like or Marley-esque moments tend to melt into the glide-and-glisten of the sound. Lyrically, Burna muses on fame, destiny, and, on “Monsters You Made,” the legacy of colonialism in Nigeria. The historical sweep of the music is equally broad, not simply focused on this-minute sounds but spanning decades of influences and collaborators-the latter ranging here from ancestral icon Youssou N’Dour to nineties legends like Timbaland and Diddy, to recent stars like Stormzy. Rhythmically, Burna’s Afro-fusion sound connects homebase Lagos to Kingston, Atlanta, and London. As befits an artist obsessed with being a superhero, Burna Boy’s music is thoroughly posthuman: much of its succulence comes from how the singer’s lilting cadences mesh with Auto-Tune. Afrobeats is a prime example of the future-pop that actually transpired, a hyper-digital sound far easier and oozier on the ear. Twentieth-century listeners imagined the music of the next millennium as a harsh, mechanistic grind or a frenzy of twitchy glitches. Listen/Buy: Rough Trade | Apple Music | Spotify | Tidal Baby can jump from whining about middle-school crushes to name-dropping denim brands to attempting to encapsulate one of the most tense and unjust moments of our lifetimes. The final addendum came in June: the protest anthem “The Bigger Picture” is a jarring inclusion on an album that isn’t overtly political-and that also makes it perfect. Instantly, it felt more like an unfocused, loose, and chaotic mid-aughts Lil Wayne mixtape with memorable tracks like “All In,” where the flexes are brilliantly batshit (Lil Baby threatens to wreck his Lambo truck just to prove, to no one in particular, that it’s not rented). But with the addition of six songs on the deluxe edition in May, the record kicked dirt on the arbitrary and outdated rules of rap albums. In February, it was a memorable yet conventional 20-track Atlanta rap album. It’s taken time for Lil Baby’s My Turn to grow into the beautiful, sprawling mess that it is. It is a certified B Corporation, a member of 1% For The Planet and certified carbon neutral. The charger is made from 72% recycled plastic and the company aims to be carbon neutral. The box includes a USB-A to USB-C cable, a getting started guide, and the 10,000mAh charger. One downside is that when you’re not using this bag, you might just throw it away. You can fill this bag with up to a pound of old electronics, which you will then send to Nimble for recycling. The only piece of plastic is a (recycled) plastic bag included in the box. Nimble takes these steps.įor starters, Nimble is skipping plastic packaging in favor of recyclable cardboard. But at least companies can take steps to minimize their impact on the environment and encourage practices like electronics recycling. It’s a tricky subject as lithium-ion will be essential to our electrified future. Lithium must be mined, which has significant environmental costs. The batteries used in phones, portable chargers, and electric cars are all lithium-ion. Tech is tough on the planet and e-waste is a big problem, especially cables and batteries. Included in the box: Portable Battery, USB-A to USB-C Cable, Getting Started Guide, Electronics Recycling Bag It’s environmentally friendly That’s why I think it’s one of the best portable chargers you can buy, especially for Android and Pixel devices. The battery is small at 3700mAh and when I run multiple apps like Maps and Spotify it also drains quickly.Īfter looking through several portable chargers, I settled on the Nimble CHAMP 10,000 mAh, which was first released in 2021. The S22 is a great phone with a sleek design and great cameras, and I love that I can actually use it with one hand. I chose the S22 base model over the more powerful versions because I wanted something smaller and more affordable and didn’t expect to need the high-end features of the S22 Ultra or the S22+. I started looking for a quality portable charger after upgrading to the Galaxy S22, my first flagship phone in a few years. You might also want to try Guayaki organic loose leaf yerba mate. I like all of the flavors, but Lemon Elation is my favorite. The flavor is quite different compared to drinking hot yerba mate with a gourd and bombilla or even compared to homemade tereré (ice-cold mate). Guayaki yerba mate energy drinks taste best ice-cold. I recommend that you try the variety packs first to find your favorite ones! You can also buy both versions in a variety pack to try different flavors. The carbonated 12 oz cans have 55 mg of caffeine per can. The carbonated ones come in 12 oz cans.Įach 15.5 oz can contains about 150mg of caffeine, which is about the same as 1 or 2 cups of coffee (depending on the type of coffee and cup size). They have a very eco-friendly mentality and are helping small farmers in creating an income while conserving the rainforest.įor the non-carbonated products, you can order a pack of 12 cans (15.5 oz each) and choose from flavors such as Lemon Elation, Enlighten Mint, Orange Exuberance, Revel Berry, and Bluephoria. Their mate is grown in Brazil with a commitment to organic and sustainable farming. Guayaki is a North American yerba mate company that has fair trade and kosher certificates. The ingredients are organic and consist of Guayaki yerba mate tea and yerba mate extract, water, sugar, and natural flavoring ingredients. They have a natural and refreshing taste. Guayaki yerba mate energy drinks come in 9 delicious flavors - 5 non-carbonated and 4 sparkling. Guayaki Yerba Mate Energy Drink – Organic and Healthy Read on to learn more about my 4 favorite yerba mate energy drinks. If you want to know more about energy drinks in general, check out this excellent book on Amazon: Are You a Monster or a Rock Star? A Guide to Energy Drinks Best Yerba Mate Energy Drinks Reviewed Or pack a few in the cooler bag for a picnic or exercise session. If you are looking for something more healthy than the usual options (Red Bull etc.) then go with yerba mate!Ĭonvenient: Just grab one from the fridge and pop it open. Refreshing: There are a few things more refreshing on a hot summer day than an ice-cold yerba mate energy drink made with natural, high-quality ingredients.Ī Healthier Choice: Yerba mate energy drinks contain plenty of antioxidants and other healthy compounds instead of artificial ingredients. Also, the effect lasts for many hours and comes down slowly without a crash. You will feel energized and focused without getting jittery. However, its effect is different compared to coffee. Lemon Balm was chosen to help reduce stress and allow you to unwind, which can be consumed before bed for a flawless rest for the mind and body.Nice Energy Boost: Yerba mate contains plenty of caffeine. We selected Gotu Kola for its ability to re-energise the brain and increase calmness. The two power-packed nootropics that feature in our Relax drink are Gotu Kola from Peru and Lemon Balm. Nootropics We carefully select and blend 100% natural nootropics from plants to help you find balance, relax, and chill out. What makes Yerba Mate the perfect fix for Go Mate Relax is that it is naturally sugar free and contains superb sleep-friendly properties! Naturally occurring magnesium and B vitamin will help you de-stress and wind down. We like to think of Yerba Mate as something of a super herb, containing a clever blend of antioxidants, vitamins, and minerals. For our Relax drink, we have removed the caffeine. Yerba Mate Hailing from South America, Yerba Mate is a traditional herbal tea made from the Yerba Mate plant. How exactly do these plant-powered marvels work in our Relax drink? Let us give you the full details. At only 12 calories per can, Relax can be enjoyed as part of a healthy, balanced lifestyle, allowing you to regain a sense of calm and balance after a busy day. A botanical blend of caffeine free Yerba Mate infusion, nootropics, and natural fruit juices, the goodness our ingredients provide speaks for themselves. You won’t find any added sugars or anything nasty in our drinks, only mind-boosting ingredients with natural calming qualities. Quench your thirst and give your tastebuds an irresistible treat with the fresh passionfruit flavour of our 100% natural relaxing caffeine free Yerba Mate and nootropics drink to help you switch off and restore your energy for when you need it. It helps you switch off and recharge at the end of a busy day. To nourish it, exercise, optimise and rest it. And as life continues to up the pace, placing more emphasis on our minds over our bodies, it has never been more important to look after our brain. Our brains consume 20% of our body’s energy. Mariah, always a pop nerd at heart, handles the alluring-but-firm vocal with gleeful aplomb, while Jimmy Jam and Terry Lewis - producers on both original and cover - go absolutely nuts on the synths in the back. “Didn’t Mean to Turn You On” ( Glitter, 2001)Īnother flashback jam, but a more literal one, as Carey takes on Cherrelle’s 1984 dance-pop gem “I Didn’t Mean to Turn You On” as her ‘80s-era Glitter avatar Billie Frank. Can’t say the victory lap wasn’t well-earned. “Forever” wasn’t released as an official single - topping the Hot 100 for a combined 26 weeks with Daydream ’s first three singles was perhaps sufficient chart dominance for one album cycle - but took over MTV anyway with its video, shot during her sold-out, fever-pitched live dates at the Tokyo Dome on the recent Daydream World Tour. Read our ranking below, and check out a Spotify playlist of the entire list at the bottom.Ī sweetly waltzing ballad of devotion, whose convincing throwback pop provided an early model for Yours Truly -era Ariana Grande. To celebrate this formidable song catalog, Billboard has compiled a list of its staff’s 100 favorite Mariah Carey songs - only including our favorite version for each individual song - which take us on a journey unlike any other in pop history, and, three decades in, still has us feeling emotions deeper than we ever dreamed of. From 1990’s debut torch song “Vision of Love” to this year’s Lauryn Hill teamup “Save the Day,” it’s been three decades of gems from the Elusive Chanteuse: songs that have soundtracked countless first loves, friendships, devastating breakups, nights out, family crises, personal epiphanies, moments of crippling loss, moments of unexpected strength - and of course, every holiday season since 1994. That includes the smash hits, naturally - many of which have become iconic - but also the fan-favorite commercial flops, the more personal deep cuts, the brilliantly curated and executed remixes. 2).īut of course, the best way to understand the legacy of Mariah Carey is still through her classic songs. 29), and The Rarities, a career-spanning compilation of B-sides and unreleased material to be unveiled on Friday (Oct. Much of that 30-plus-year history will be more fully explained by two much-anticipated projects Carey is releasing this week: The Meaning of Mariah Carey, her revealing memoir co-written with Michaela Angela Davis that dropped on Tuesday (Sept. Of course, all of these stats and achievements only scratch the surface on the full Mariah Carey story - one that also includes several moments of game-changing collaboration, multiple generations of enduring artistic influence, more dramatic turns than one of her action-packed videos and a wider scope of highs and lows than even her five-octave vocal range. Let's step back a minute and look at the history of the equal sign. It is intolerable to have two different values on each side of an equals sign. I say western children as research from Ma (1999) cited in Darr (2001) says “From the Chinese teachers’ perspective … the semantics of mathematical operations should be represented rigorously. Using the appropriate symbol to distinguish definition statements from equality statements may go a long way, at least in proportion to the effort of using them, towards alleviating confusion in students of math and computer science.When you see an equal sign what do you think it means?Ī lot of research of western educated children shows that many don't see it as meaning 'is the same as' but see it more as the thing that comes before the 'big' number in a sentence. That is, $X := Y$ means “use $X$ as a symbol for $Y$”, which differs from “use $Y$ as a symbol for $X$.” In contrast, the standard equals sign “=” is appropriately symmetric. I prefer this symbol over the popular “$\equiv$” symbol because it emphasizes the assymetry of the statement. I think it’s important to use the symbol “:=” to denote definition. Use “:=” instead of “=” to denote definition Though I have not seen any data on the topic, I wonder whether teaching these two operators from the very beginning of a student’s mathematical education would alleviate this common confusion. In contrast, the statement x = y returns either True or False depending on whether the value referenced by x is equal to the value referenced by y. That is, the statement x = y assigns the value referenced by symbol y to symbol x. That is, that we can create this definition at all! Overloading the equal sign creates confusion in computer programmingĪnyone who has taught introductory computer programming is familiar with the very common confusion between the assignment operator and equality operator in programming languages.įor example, in many programming languages, like C and Python, the assigment operator uses the standard equals sign. The real interesting quality to this definition is that the ratio of the sides of a right triangle are a function of its angles regardless of the lengths of the sides. The statement is defining $\sin \theta$ to be the ratio between the opposite side to the hypotenuse. The mystery was, at least partly, alleviated by the clarification that $\sin \theta$ is not an object that existed before we saw this statement – rather, this statement created the object for the first time. Their confusion arose from the erroneous interpretation of this statement as describing an equality rather than a definition. Their question was more along the lines of, “What is this mysterious thing? And why on earth is it equal to the ratio of the sides of the triangle?” It wasn’t, “Why are the ratios between the sides of a right-triangle functions of the angles between those sides?” Nor, “Why is this definition important?” Rather, their confusion seemed to stem from the very existence of this mysterious object, “$\sin \theta$”. Given some entity denoted with the symbol $Y$, the statement “let $X$ be $Y$”, also often denoted $X = Y$, means that one should use the symbol “$X$” to refer to the entity referred to by “$Y$”.įor example, in introductory math textbooks it is common to define the sine function in reference to a right-triangle:\[\sin \theta = \frac$?” They never explicitly stated so, but it become evident that their confusion was not the good kind of confusion. Said differently, the quantity $c^2$ is the same quantity as the quantity $a^2 + b^2$. The Pythagorean Theorem says that $a^2 + b^2 = c^2$. The statement “$X$ equals $Y$”, denoted $X = Y$, means that $X$ and $Y$ are the same thing.įor example, let’s say we have a right-triangle with edge lengths $a$, $b$ and $c$, where $c$ is the hypotenuse. Let’s say we have two entities, which we will denote using the symbols $X$ and $Y$. To ensure that we’re on the same page, let’s first define these two notions. Early in my learning days, I believe that this overloading of the equal sign led to more confusion than necessary and I have personally witnessed it confuse students. I find it unfortunate that two of the most important relationships in mathematics, namely equality and definition, are often denoted using the exact same symbol – namely, the equal sign: “=”. In this post, I argue for the use of two different symbols for these two fundamentally different operators. The overloading of this symbol confuses students in mathematics and computer programming. Two of the most important relationships in mathematics, namely equality and definition, are both denoted using the same symbol – namely, the equals sign. |
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