One Hundred

Historians think numbers began with the words one and two.

Me You. Man Woman. One Two.

One Two are different from all the numbers that follow. They indicate duality. A pair. By themselves they don't necessarily imply a counting sequence.

Then Three came along (children?).

One Two Three. Me You Others. Man Woman Many. Dad, Mom, kids.

At some point we figured out how to count. We needed a specific new word beyond Many. It was Four.

(By the way, not all cultures count. Some recognize things and keep track without assigning numbers.)

Once we had a whole big bunch of things (family) around us, we needed a new unit. A Bigger ONE. In English the new "big" one is called TEN.

We've added this new name TEN, and we continue to treat the One Two that follow TEN in a unique way - they are different from all the rest. The number value is ten plus one and ten plus two but in English we say Eleven and Twelve. Two completely new words.

Then we say Thirteen (three and ten). All the following numbers use that form. We didn't invent new words, but use a combination of old words. Using the structure of four and ten, etc. Fourteen, Fiveteen, Sixteen, Seventeen, Eighteen, Nineteen.

Once we needed yet another set of TEN, we completely gave up on this "teen" structure. We didn't invent more new words but we started re-assembling words in a different sequence - twenty-and-one, twenty-two, etc. This goes all the way up to the next BIG TEN.

We reached a threshold. We needed a new number. A much bigger ONE. A much bigger TEN. We call it One Hundred.

One hundred. A nice round sound. I like that number.

NOTE: If this kind of thing interests you, you may want to look at the book Number Words and Number Symbols, A Cultural History of Numbers, by Karl Menninger.

I mention all this history of numbers, words and groups, thresholds and units because yesterday I noticed that this blog has been visited by math-loving people from 100 countries. Thanks!

PS - and then the counter software broke. Whoops... I'll try to get it back.

Fun Food Math, Part 5: Color

OK, some of these blogs on food math haven't been much fun. I admit when trying to get the numbers straight we often forget the word fun.

So let's see if we can fix that today. I can easily eat brown, brown, brown. Like a bowl of lentil and rice soup with pita bread and some dates (today's lunch). Or brown and beige for fish and chips.

But I prefer colorful food. (Is color fun?)

Take a look at the dishes in the tall picture and decide what math we could do with them:

1. Could you put the ingredients into categories (Meat, Fish, Vegetables) like the chart yesterday?  
  Yes.

2. Could you estimate the number of ingredients in each dish?  
  Maybe. I could do it because I cooked these dishes myself and can remember most of the ingredients.

3. Perhaps you might calculate the caloric energy in a serving for each menu item?  
  Wow - that would be very hard. But possible with the right data sources.
 

4. Could you determine the cost of the food in each of these pictures?  
  That would take a lot of facts. I guess I could estimate what they cost.

5. Could you count the colors in each of the food pictures?
  No that's impossible!

6. Could you figure out what season it was when we made each meal?  
  Wow - that would be very difficult, given the way food is flown around the world...

7. Could you calculate how many miles the ingredients traveled to get to my plates?
   Next to impossible, I'd think. However we tend to use fairly local foods when we can.

8. Could you calculate the relative amounts of vegetable versus meat/fish in each dish?
  Yes, that should be pretty easy. I see cheese, eggs, fish, scallops and pork in these dishes,
  plus mushrooms, vegetables, fruits, pasta and bread.

9. Can you calculate how much salt is in our diet from these photos?
  Not really, but you can see my favorite salt and pepper shaker which I've had for 35 years. If you never eat processed food, you won't get that much salt.

10. Maybe we could take a break, have lunch or dinner, and ponder the math implications of meals? Yes, sounds good. By the way, a handful of salad and grapefruit helped make my lunch more colorful.

(Yes, the same carrots and lettuce were in yesterday's picture too. But they have been eaten now.)

Fun Food Math, Part 4: Labels

Math and food seem to go together in our post-industrial age, especially on our food labels.

And how did I know researching and producing this page 2 days ago, that the US would slap the hands of 17 companies for their deficient or misleading labels!

In the European Union, Nutritional Labels must indicate a food's 

• energy value
• amount of protein, carbohydrate, sugar, fat, saturated fatty acids, dietary fibre and sodium

The energy and nutrients are given per 100g or per 100ml. In addition, they can be expressed per package or per portion. Vitamins and minerals must be shown as a percentage of the recommended daily allowance (RDA). Nutrition labelling may also include  mono- and poly-unsaturated fatty acids, cholesterol and certain mineral salts and vitamins.

This information must be placed together, in a clearly visible place on the package, in legible, indelible characters and expressed in easily understood words in the local language of the buyer.

Despite all the hard work on numbers and facts, people are still very confused. The Europeans are now considering the stop light approach - red bad, orange fair, green good. Here's an example from the UK:

This label was much more effective than the typical ones in helping people decide the healthfulness of food. Red - danger - don't buy. I can see how the food producers don't like that, but then maybe they should mix up ingredients in a healthier way??


The USA's labeling requirements are long, complicated and amusing to read (discussions on the weight of one cherry, variations in size of fish and pickles, do-it-yourself pizza kits, etc.) The result is about the same as in the EU, a complicated label revealing what the food contains. Lots of numbers.

I found some nifty software programs aimed at food producers. They allow you to enter all your food's ingredients, then generate the appropriate labels for the regions where you want to sell. They save lots of math and typesetting hassles, I'm sure. And there are companies that audit your labels to keep you IN compliance and OUT of trouble. Why are these services needed?

 Many labels are likely to be wrong. But that doesn't necessarily mean they are wrong in the healthy direction ...

You can get nutritional data yourself from the USDA's food database site.  Choose the first search option and enter your choices.

Government agencies spend lots of their time and our money figuring out what food is good for us:

The mission of the Human Nutrition Research Center is to research food ... optimizing human health and reducing the risk of nutritionally related disorders ... how dietary components interact with genetic, physiological, sociological, and environmental factors and affect the health of the American population ... 

You probably grew up with a food quality research director in your own house: Your Mom.

If she's not around, use your forefinger instead. Put it over the label. If you can cover the whole list of ingredients with your finger, then buy the food. If the list of ingredients is longer than the width of your finger,

 

PUT THE PACKAGE BACK!

GRAB SOMETHING ELSE INSTEAD.

Fun Food Math, Part 3: Calories

Calories! The number everyone hates to count.

But what is a calorie? Google helpfully suggests that it's equal to 4184 joules. That doesn't help us much!

A calorie is a unit of measure of heat change. It's an obsolete, archaic unit. A confusing term, too. If we give the various calorie units their proper names, we've got:

• Kilogram calorie - heat required to raise the temperature of a kilogram of water one degree Celsius.
• Gram calorie - heat required to raise the temperature of a gram of water one degree Celsius.
• Pound calorie - heat required to raise the temperature of a pound of water one degree Celsius.
• British Thermal Unit (BTU) - heat required to raise the temperature of one pound of water one degree Fahrenheit. It's roughly equal to the energy released when one wooden match is burned down to the very end. 

All of these units have been superseded by the joule, which is a metric unit of energy. A joule in body heat terms is roughly the amount of heat radiated by a human being every 100th of a second.

But I've strayed from fun food math. We want to know about CALORIES that make us fat!

The calorie that makes us fat is a unit of measure describing how our cells convert food to energy through a process called respiration. The cells combine the food with oxygen, which releases energy.
 
Think about this - a pound of lettuce will be harder to burn and produce less energy than a pound of cheese, so lettuce has fewer potential calories.

Are calories determined by burning some food in a dish (like the match) and measuring the output?

NO. For many reasons:

1. Technically calories aren't IN food, they are the result of a reaction that happens when food is processed through respiration in our cells. Because this process isn't 100% efficient, scientists have decided to "discount" the potential energy in a quantity of food by 15%. The actual intake of 100 calories of food results in 85 calories of usable energy. We only count the 85.

2. There's another discount applied to calories, because many foods contain indigestable fiber. Our bodies don't digest the fiber and so that potential energy is passed through the body and lost. If we burned the food in a dish the fiber would burn too, and influence the results. Thus food scientists discount the caloric content of fibrous foods.

3. The process of doing these conversions is complex and requires more chemical ability than most people have. There are reference books and databases from which everyone draws their caloric information.

So the calories we eat are used up by the muscles, right?  Yes and No.

1. The potential calories we generate get converted into a substance called ATP that is passed from cell to cell (imagine an energy traveler's check that any cell will accept, but each party has to pay fees when they buy or sell).

The conversion cost with ATP is pretty high, so only about 18-26% of the calories freed by respiration actually get used in the muscles to produce energy (running, bicycling, scooping guacamole with a tortilla chip, etc.)

2. Anything not used quickly by the muscles gets converted into fat (which is like putting your traveler's check into a bank account rather than taking cash). Fat is our reserve from which the body draws between meals. And there's a service charge for converting into and out of fat as well, like changing to a different country's currency.

We'll have to save nutritional labels and calorie counting for another day.

Fun Food Math, Part 2: Money

Yesterday we talked about chocolate. Today we look at the intersection of food and money. Here's the question:

 How MUCH do YOU spend on food?

Fish and Chips 

Do you keep track of every penny your family spends on food? We don't. I'm not sure how much we spend, although I am sure my wife could tell me what she buys and brings home. Each of us spends some money on food at work or school.

We had a little survey around Excel Math and most of us knew our weekly grocery spending, but not what the totals were for a month, including dining out, buying food gifts, food for the pantry, coffee, snacks, etc.

Most governments monitor how its citizens spend their funds on food. In some cases it's to ensure food assistance programs are adequately funded. With foods such as corn, they want to know how much goes to animals, becomes corn syrup, gets turned into ethanol, or is eaten directly by people.

USA FOOD NUMBERS

Here's one government report, from about 5 years ago. During this same time period, median household incomes varied from $35-60,000 (depending on the state).

Average spending per month in dollars for a household of four, 2004-05
Food spending	Low- income $10k-29k	Mid- income $30k-$49	High-income $50k +	All incomes
Total food spending Food Away From Home Food At Home Meat/poultry/fish and seafood Fruit and vegetables Cereals/bakery products Dairy products Sugars/sweets Fats/oils Nonalcoholic beverages Miscellaneous	462 129 334 91 54 46 38 11 10 30 52	527 195 332 86 50 46 40 10 10 31 56	816 374 441 105 71 61 51 17 11 40 81	700 298 402 100 64 56 46 15 10 37 72
Note: Totals may be off due to rounding. Miscellaneous includes frozen/canned meals and soups; chips, nuts, and snacks; condiments, etc.      Source: Consumer Expenditures Survey, Bureau of Labor Statistics.

The first row (in bold) is the total of everything below it. They totalled upwards, I guess they assumed we wouldn't read all the way to the bottom. The second row shows the expenditures on Food Away from Home, and Food At Home is the total of the food types shown beneath it.

As the note at the bottom indicates, the totals don't add exactly due to rounding at various stages of data collection. It looks like each of the columns is off by 2-3. How much variation in totals is normal, or acceptable? I don't know.

The miscellaneous row includes lots of items that are certainly part of the rows above - such as frozen dinners which are difficult to categorize but likely to contain both meat and vegetables.

INTERNATIONAL FOOD NUMBERS

On average, US families spend a bit less than 10% of their Private Consumption Expenditures on food. In the UK it's 11%, in France 16%, Italy 25%, etc. This little table shows the average values for food expenditures, as a percentage of family spending:

Percentage of Family Spending on Food
16%       High-income countries
35%       Mid-income countries
55%       Low-income countries

There are lots of political and economic consequences that flow outward from this state of inequality. But that's well beyond the scope of our blog today.

Figs and Cheese and Salami 

MATH IMPLICATIONS

What have we learned so far? Food spending ranges from 10-55% of total family spending. The more money you have, the more you can spend on food, but there seems to be a limit - food spending does not increase infinitely.

We have to trust that we and our fellow citizens knew what we spent and fairly reported those expenses when asked about it. We trust that government agencies kept all the numbers in pristine condition - when the numbers were categorized, processed, summed, averaged and interpreted - nothing got lost.

We trust that the person who creates these tables gets the rows and columns right! (Or if there were errors, enough people worked at this subject separately that their errors were insignificant or obvious enough to get fixed).

These cautions hold true in ANY math endeavor you undertake.

I think it's fair to say the bigger the collection of numbers, the more carefully we must handle them and the more likely it is that they are untrustworthy. Statistics is the business of sorting out collections of numbers. We just barely touch on it in elementary school math. We're busy with other things...

Brownies, Ice Cream, Nuts and Sauce, with a Cherry on top!

Fun Food Math, Part 1: Chocolate

This past week I have been reading about Malcolm Muggeridge, an English journalist for much of the twentieth century. His diaries tell of living in Moscow in 1932. One line he wrote stuck in my mind,

Mr C proudly told me he ate his weight in chocolate every year ...

Well. That means hundreds of pounds of chocolate a year. Let's say he weighed 183 lbs - that would be half a pound of chocolate a day! Obviously Mr C didn't have a problem with guilt. Or his doctor. Or perhaps he didn't care. When I described this to my wife, a real chocolate lover, she said,

That's got to be 1000 calories a day!

I decided to look it up. Modern chocolate is around 150-160 calories per ounce!

150 x 8 = 1200 calories per half-pound.

That's about half the total recommended caloric intake per day for an adult, but then we don't know exactly how large Mr. C was, or how much work or exercise he did.

In my research I discovered there are many kinds of chocolate. We're going to assume Mr. C ate dark European chocolate, because of the time and place where he lived. Let's see what the nutritional components are for chocolate. The values shown are possible percentages by weight, not percentages of your daily recommended intake.

Chocolate Nutritional Details

Ingredient	Dark Euro	Semi-Sweet
Fat	10-15%	20-35%
Carbs	45-60%	50-70%
Sugar	2%	45-65%
Fiber	20-35%	3-8%
Protein	17-22%	3-8%

Notice these numbers do not add to 100%. Hmmmm

I noticed one interesting fact while researching this - European chocolate is only 60 calories per ounce. That means his intake of calories could have been much less back then compared to eating chocolate today:

60 x 8 = 480 calories per half-pound.

In addition to his calories, Mr. C was getting anti-oxidants, theobromine, phenylethylamine, caffeine, serotonine, vitamins and minerals.

The fat he was eating was from cocoa butter, which is made up of stearic acid, oleic acid, and palmitic acid. All of these are basically neutral as far as cholesterol levels go, as they are plant and not animal-based fats. Here's a look at the makeup of a cocoa bean, which is used to make chocolate. This comes from Hershey.

If you like chocolate, you could eat a dark chocolate most often, and use sweetened, milk and white chocolates as treats, NOT a primary food group! :-)

 

PS: No source recommends eating the equivalent 

of your weight each year of any single food!

Revised  3-2-2010