HOW ARE MATRICES USED IN REAL LIFE

In everyday applications, matrices are used to represent real-world data, such as the traits and habits of a certain population. They are used in geology to measure seismic waves. Matrices are rectangular arrangements of expressions, numbers and symbols that are arranged in columns and rows. Matrices have "m" number of rows and "n" number of columns, and numbers in a matrix are called entities or entries.

Matrices are common tools used by the science and research industry to track, record and display the results of research. In addition to applied science, matrices are also used in the basic sciences. For example, physicists use matrices to study optics, electrical circuits and quantum mechanics. The discipline of physics also uses matrices to calculate battery power outputs and resistor conversion of electrical energy into a more efficient form. Matrices are used to solve problems involving Kirchoff's laws of voltage and current. Computer science also relies heavily on matrices. Tasks such as projecting a three-dimensional image onto a two-dimensional screen and encrypting message codes are two areas in which matrices are used. Matrices and their inverses are necessary for programmers to code encrypted messages, and these codes are used to run Internet programs, such as search systems.

MATHS TRICK IN MULTIPLICATION

1. Multiplying by 9, or 99, or 999

Multiplying by 9 is really multiplying by 10-1.

So, 9x9 is just 9x(10-1) which is 9x10-9 which is 90-9 or 81.

Let's try a harder example: 46x9 = 46x10-46 = 460-46 = 414.

One more example: 68x9 = 680-68 = 612.

To multiply by 99, you multiply by 100-1.

So, 46x99 = 46x(100-1) = 4600-46 = 4554.

Multiplying by 999 is similar to multiplying by 9 and by 99.

38x999 = 38x(1000-1) = 38000-38 = 37962.

2. Multiplying by 11

To multiply a number by 11 you add pairs of numbers next to each other, except for the numbers on the edges.

Let me illustrate:

To multiply 436 by 11 go from right to left.

First write down the 6 then add 6 to its neighbor on the left, 3, to get 9.

Write down 9 to the left of 6.

Then add 4 to 3 to get 7. Write down 7.

Then, write down the leftmost digit, 4.

So, 436x11 = is 4796.

Let's do another example: 3254x11.

The answer comes from these sums and edge numbers: (3)(3+2)(2+5)(5+4)(4) = 35794.

One more example, this one involving carrying: 4657x11.

Write down the sums and edge numbers: (4)(4+6)(6+5)(5+7)(7).

Going from right to left we write down 7.

Then we notice that 5+7=12.

So we write down 2 and carry the 1.

6+5 = 11, plus the 1 we carried = 12.

So, we write down the 2 and carry the 1.

4+6 = 10, plus the 1 we carried = 11.

So, we write down the 1 and carry the 1.

To the leftmost digit, 4, we add the 1 we carried.

So, 4657x11 = 51227 .

3. Multiplying by 5, 25, or 125

Multiplying by 5 is just multiplying by 10 and then dividing by 2. Note: To multiply by 10 just add a 0 to the end of the number.

12x5 = (12x10)/2 = 120/2 = 60.

Another example: 64x5 = 640/2 = 320.

And, 4286x5 = 42860/2 = 21430.

To multiply by 25 you multiply by 100 (just add two 0's to the end of the number) then divide by 4, since 100 = 25x4. Note: to divide by 4 your can just divide by 2 twice, since 2x2 = 4.

64x25 = 6400/4 = 3200/2 = 1600.

58x25 = 5800/4 = 2900/2 = 1450.

To multiply by 125, you multipy by 1000 then divide by 8 since 8x125 = 1000. Notice that 8 = 2x2x2. So, to divide by 1000 add three 0's to the number and divide by 2 three times.

32x125 = 32000/8 = 16000/4 = 8000/2 = 4000.

48x125 = 48000/8 = 24000/4 = 12000/2 = 6000.

4. Multiplying together two numbers that differ by a small even number

This trick only works if you've memorized or can quickly calculate the squares of numbers. If you're able to memorize some squares and use the tricks described later for some kinds of numbers you'll be able to quickly multiply together many pairs of numbers that differ by 2, or 4, or 6.

Let's say you want to calculate 12x14.

When two numbers differ by two their product is always the square of the number in between them minus 1.

12x14 = (13x13)-1 = 168.

16x18 = (17x17)-1 = 288.

99x101 = (100x100)-1 = 10000-1 = 9999

If two numbers differ by 4 then their product is the square of the number in the middle (the average of the two numbers) minus 4.

11x15 = (13x13)-4 = 169-4 = 165.

13x17 = (15x15)-4 = 225-4 = 221.

If the two numbers differ by 6 then their product is the square of their average minus 9.

12x18 = (15x15)-9 = 216.

17x23 = (20x20)-9 = 391.

5. Squaring 2-digit numbers that end in 5

If a number ends in 5 then its square always ends in 25. To get the rest of the product take the left digit and multiply it by one more than itself.

35x35 ends in 25. We get the rest of the product by multiplying 3 by one more than 3. So, 3x4 = 12 and that's the rest of the product. Thus, 35x35 = 1225.

To calculate 65x65, notice that 6x7 = 42 and write down 4225 as the answer.

85x85: Calculate 8x9 = 72 and write down 7225.

6. Multiplying together 2-digit numbers where the first digits are the same and the last digits sum to 10

Let's say you want to multiply 42 by 48. You notice that the first digit is 4 in both cases. You also notice that the other digits, 2 and 8, sum to 10. You can then use this trick: multiply the first digit by one more than itself to get the first part of the answer and multiply the last digits together to get the second (right) part of the answer.

An illustration is in order:

To calculate 42x48: Multiply 4 by 4+1. So, 4x5 = 20. Write down 20.

Multiply together the last digits: 2x8 = 16. Write down 16.

The product of 42 and 48 is thus 2016.

Notice that for this particular example you could also have noticed that 42 and 48 differ by 6 and have applied technique number 4.

Another example: 64x66. 6x7 = 42. 4x6 = 24. The product is 4224.

A final example: 86x84. 8x9 = 72. 6x4 = 24. The product is 7224

7. Squaring other 2-digit numbers

Let's say you want to square 58. Square each digit and write a partial answer. 5x5 = 25. 8x8 = 64. Write down 2564 to start. Then, multiply the two digits of the number you're squaring together, 5x8=40.

Double this product: 40x2=80, then add a 0 to it, getting 800.

Add 800 to 2564 to get 3364.

This is pretty complicated so let's do more examples.

32x32. The first part of the answer comes from squaring 3 and 2.

3x3=9. 2x2 = 4. Write down 0904. Notice the extra zeros. It's important that every square in the partial product have two digits.

Multiply the digits, 2 and 3, together and double the whole thing. 2x3x2 = 12.

Add a zero to get 120. Add 120 to the partial product, 0904, and we get 1024.

56x56. The partial product comes from 5x5 and 6x6. Write down 2536.

5x6x2 = 60. Add a zero to get 600.

56x56 = 2536+600 = 3136.

One more example: 67x67. Write down 3649 as the partial product.

6x7x2 = 42x2 = 84. Add a zero to get 840.

67x67=3649+840 = 4489.

8. Multiplying by doubling and halving

There are cases when you're multiplying two numbers together and one of the numbers is even. In this case you can divide that number by two and multiply the other number by 2. You can do this over and over until you get to multiplication this is easy for you to do.

Let's say you want to multiply 14 by 16. You can do this:

14x16 = 28x8 = 56x4 = 112x2 = 224.

Another example: 12x15 = 6x30 = 6x3 with a 0 at the end so it's 180.

48x17 = 24x34 = 12x68 = 6x136 = 3x272 = 816. (Being able to calculate that 3x27 = 81 in your head is very helpful for this problem.)

9. Multiplying by a power of 2

To multiply a number by 2, 4, 8, 16, 32, or some other power of 2 just keep doubling the product as many times as necessary. If you want to multiply by 16 then double the number 4 times since 16 = 2x2x2x2.

15x16: 15x2 = 30. 30x2 = 60. 60x2 = 120. 120x2 = 240.
23x8: 23x2 = 46. 46x2 = 92. 92x2 = 184.
54x8: 54x2 = 108. 108x2 = 216. 216x2 = 432.

Practice these tricks and you'll get good at solving many different kinds of arithmetic problems in your head, or at least quickly on paper. 

EVERYDAY EXAMPLES OF MATH IN THE REAL WORLD

When am I ever going to use this?

Variations of this question have echoed through the halls of math classrooms everywhere. Struggling students often become frustrated with complex math problems and quickly give in to the notion that they will never use math in “real life” situations. While it may be true that some of the more abstract mathematical concepts rarely come into play, the underlying skills developed in high school math classrooms resonate throughout a student’s lifetime and often resurface to help solve various real world or work-related problems sometimes years down the line.

Ask any contractor or construction worker and they’ll tell you just how important math is when it comes to building anything. Creating something that will last and add value to your home out of raw materials requires creativity, the right set of tools, and a broad range of mathematics.

Figuring the total amount of bags of concrete needed for a slab, accurately measuring lengths, widths, and angles, and estimating project costs are just a few of the many cases in which math is necessary in real life home improvement projects.

Some students may say they don’t plan on working in construction and this may be true, but many will own a home at some point in their life. Having the ability to do minor home improvements will save a lot of money and headache. Armed with math, they will also have the ability to check the work and project estimates, ensuring they’re getting the best value.

One of the more obvious places to find people using math in everyday life is at your neighborhood grocery store. Grocery shopping requires a broad range of math knowledge from multiplication to estimation and percentages.

Calculating price per unit, weighing produce, figuring percentage discounts, and estimating the final price are all great ways to include the whole family in the shopping experience.

Teacher Tip: Encourage your students to play math challenges at the grocery store with their family by attempting to estimate the total cost of all groceries prior to checkout. The difficulty can be increased by incorporating coupons, sales, and adjusted pricing for bulk items. Your little bargain shoppers will thank you later when they’re saving money on their own groceries.

More math can be found in the kitchen than anywhere else in the house. Cooking and baking are sciences all their own and can be some of the most rewarding (and delicious) ways of introducing children to mathematics. After all, recipes are really just mathematical algorithms or self-contained step-by-step sets of operations to be performed. The proof is in the pudding!

Working in the kitchen requires a wide range of mathematical knowledge, including but not limited to:

• measuring ingredients to follow a recipe
• multiplying / dividing fractions to account for more or less than a single batch
• converting a recipe from Celsius to Fahrenheit
• converting a recipe from metric (mL) to US standard units (teaspoon, tablespoon, cups)
• calculating cooking time per each item and adjusting accordingly
• calculating pounds per hour of required cooking time
• understanding ratios and proportions, particularly in baking (ex. the recipe calls for 1 egg and 2 cups of flour, then the ratio of eggs to flour is 1:2).

Following a recipe can sometimes be tricky, especially if conversions are necessary. We Americans follow our own set of rules when it comes to most forms of measurement. Conversions make it a bit more difficult to follow recipes from other countries as they most likely use Celsius and the metric system.

Celsius to Fahrenheit Conversion

Ex. The recipe calls for the oven to be set at 220°C, but yours is labelled by Fahrenheit.

Formula: °C  x  9/5 + 32 = °F

220 x 9/5 + 32 = °F

396 + 32 = 428°F

Metric to US Standard Unit Conversion

1 US legal cup = 240 mL

1 US tablespoon = 14.79 mL

1 US teaspoon = 4.92 mL

1 US fluid ounce = 29.57 mL

Math comes in handy when travelling and shows up in various ways from estimating the amount of fuel you’ll need to planning out a trip based on miles per hour and distance traveled. Calculating fuel usage is crucial to long distance travel. Without it, you may find yourself stranded without gas or on the road for much longer than anticipated. You may also use math throughout the trip by paying for tolls, counting exit numbers, checking tire pressure, etc.

Long before GPS and Google Maps, people used atlases, paper road maps, road signs, or asked for directions in order to navigate throughout the country’s highways and byways. Reading a map is almost a lost art, but requires just a little time, orientation, and some basic math fundamentals. Teaching students how to use their math skills to read maps will make them safer travelers and less dependent on technology.

In order to use any map, you must first orient yourself, meaning to find your current position on the map. This will be point A. The simplest way to do this is to locate the town you’re in then the nearby crossroads, intersection or an easily identifiable point such as a bridge, building, or highway entrance. Once you’ve established a starting point, locate where on you want to go (point B). Now you can determine the best route depending on terrain, speed limit, etc.

Many experts agree that without strong math skills, people tend to invest, save, or spend money based on their emotions. To add to this dilemma, those individuals with poor math fundamentals typically make greater financial mistakes like underestimating how quickly interest accumulates. A student who thoroughly grasps the concepts of exponential growth and compound interest will be more inclined to better manage debt.

Financial knowledge decays over time, so it’s important to keep young people involved. By continually showing how specific math lessons apply to real life financial situations and budgeting, kids can learn how to properly spend and save their money without fear or frustration.

Time is our most valuable asset. Without proper planning, the day can slip through out fingers and our list of duties and responsibilities can start to accumulate. In our fast-paced, modern world, we can easily fall behind and get overwhelmed with all that we have to do. Keeping on schedule has greater weight in our daily lives than ever in history, but it takes more math skills than simply reading a clock or following a calendar to stay on top of everything.

WHO INVENTED ZERO?

Though humans have always understood the concept of nothing or having nothing, the concept of zero is relatively new — it only fully developed in the fifth century A.D. Before then, mathematicians struggled to perform the simplest arithmetic calculations. Today, zero — both as a symbol (or numeral) and a concept meaning the absence of any quantity — allows us to perform calculus, do complicated equations, and to have invented computers.

Early history: Angled wedges

Zero was invented independently by the Babylonians, Mayans and Indians (although some researchers say the Indian number system was influenced by the Babylonians). The Babylonians got their number system from the Sumerians, the first people in the world to develop a counting system. Developed 4,000 to 5,000 years ago, the Sumerian system was positional — the value of a symbol depended on its position relative to other symbols. Robert Kaplan, author of "The Nothing That Is: A Natural History of Zero," suggests that an ancestor to the placeholder zero may have been a pair of angled wedges used to represent an empty number column. However, Charles Seife, author of "Zero: The Biography of a Dangerous Idea," disagrees that the wedges represented a placeholder.

The Sumerians’ system passed through the Akkadian Empire to the Babylonians around 300 B.C. There, scholars agree, a symbol appeared that was clearly a placeholder — a way to tell 10 from 100 or to signify that in the number 2,025, there is no number in the hundreds column. Initially, the Babylonians left an empty space in their cuneiform number system, but when that became confusing, they added a symbol — double angled wedges — to represent the empty column. However, they never developed the idea of zero as a number.

Zero in the Americas

Six hundred years later and 12,000 miles from Babylon, the Mayans developed zero as a placeholder around A.D. 350 and used it to denote a placeholder in their elaborate calendar systems. Despite being highly skilled mathematicians, the Mayans never used zero in equations, however. Kaplan describes the Mayan invention of zero as the “most striking example of the zero being devised wholly from scratch.”

India: Where zero became a number

Some scholars assert that the Babylonian concept wove its way down to India, but others give the Indians credit for developing zero independently.

The concept of zero first appeared in India around A.D. 458. Mathematical equations were spelled out or spoken in poetry or chants rather than symbols. Different words symbolized zero, or nothing, such as "void," "sky" or "space." In 628, a Hindu astronomer and mathematician named Brahmagupta developed a symbol for zero — a dot underneath numbers. He also developed mathematical operations using zero, wrote rules for reaching zero through addition and subtraction, and the results of using zero in equations. This was the first time in the world that zero was recognized as a number of its own, as both an idea and a symbol.

From the Middle East to Wall Street

Over the next few centuries, the concept of zero caught on in China and the Middle East. According to Nils-Bertil Wallin of YaleGlobal, by A.D. 773, zero reached Baghdad where it became part of the Arabic number system, which is based upon the Indian system.

A Persian mathematician, Mohammed ibn-Musa al-Khowarizmi, suggested that a little circle should be used in calculations if no number appeared in the tens place. The Arabs called this circle "sifr," or "empty." Zero was crucial to al-Khowarizmi, who used it to invent algebra in the ninth century. Al-Khowarizmi also developed quick methods for multiplying and dividing numbers, which are known as algorithms — a corruption of his name. 

Zero found its way to Europe through the Moorish conquest of Spain and was further developed by Italian mathematician Fibonacci, who used it to do equations without an abacus, then the most prevalent tool for doing arithmetic. This development was highly popular among merchants, who used Fibonacci’s equations involving zero to balance their books.

Wallin points out that the Italian government was suspicious of Arabic numbers and outlawed the use of zero. Merchants continued to use it illegally and secretively, and the Arabic word for zero, "sifr," brought about the word “cipher,” which not only means a numeric character, but also came to mean "code."  

By the 1600s, zero was used fairly widely throughout Europe. It was fundamental in Rene Descartes’ Cartesian coordinate system and in Sir Isaac Newton’s and Gottfried Wilhem Liebniz’s developments of calculus. Calculus paved the way for physics, engineering, computers, and much of financial and economic theory.