Increased - that's a plus or multiply?

If you have a question: "Increased - is that a plus or multiply?", then definitely no answer to it. You need to read or listen to what's next. If "increased by as much as something something", then this addition and the plus sign. For example, the number 7 is increased by 2 units. What are we doing? To previous value we added increase and get new value:

7 + 2 = 9

Increased

If "increased in so many times," it is multiplication. For example, the number 7 is increased in 2 times. It is necessary to multiply the previous value of the specified number of times, and we get a new meaning:

7 * 2 = 14

As you can see, the numbers in the above examples are the same, and the result is different. It all depends on how to formulate a thought.

Snail on a pole

Objective: Snail crawling on a pole height of 10 meters. During the day, she rises to 4 meters per night descends to 3 meters. For some time the snail reaches the top of the column?

Snail on a pole

To solve this problem about a snail on a post need to revise some of its condition. The day the snail takes off the pole to a height of 4 meters. The post has a height of 10 meters. Reformulate the question: at what height should be a snail to a running start to fly to the top of the column? The answer to this question is easy to calculate:

10 - 4 = 6 meters

Now the second question: how many days should snail fidgeting up and down the pole to reach a height of 6 meters. Again, take the hands of the math and think:

6 / (4 - 3) = 6 days

As a result, we have six days cochlea will need to enter the starting position for a breakthrough victory, and one day for the race victory. The result is that the triumph of a snail on a post we will see after 7 days.

The same mathematical problem can be formulated with a modern twist. Look how it will sound in the Russian mathematical textbooks soon: From the border to Russia at the borders of NATO distance of 10 thousand kilometers. Green men move per day from the Russian border to NATO at the border at a distance of 4000 kilometers. During the night, NATO troops cast of unknown green men on the 3,000 kilometers back to Russia. How many days disappear NATO? The answer to this problem is the same: after 7 days of the great light of the Russian world illuminate the whole of Europe. By the way, the light of the Christian faith had once lit up Europe funeral pyres of the Inquisition. And the Third Reich sprawling across Europe recently. We must not forget history.

9 multiply 7 equals

Immediately write a ready answer: 9 multiply 7 equals 63. So we have the multiplication table says that we all have been taught, but to the end, many of us have not learned. I'm not an exception. Natural laziness and chronic multiple sclerosis did not allow me to remember this vital piece of mathematics. But it is knowledge of mathematics allows me to easily cope with this column multiplication tables. I just calculate the result I needed.

We all know that when multiplied by a permutation of the factors does not change the result. If somewhere sometime mathematics will tell you otherwise, do not believe them - they are wrong. Simple math multiplication like to call it, that in fact is not a multiplication. For example, if we call a cow "Multiplication", it then becomes to turn milk and poop in the hay. After all, the cow to the mathematical action has nothing to do.

So, I always use the following equation:

9*7 = 7*9

Next is quite simple. That number, which must be multiplied by 9, I multiply by 10 and take away his once:

7*9 = 7*10-7 = 70-7 = 63


Multiplied by 10 is very simple - have added to the number of the toe and the result is ready. Subtract this number from a small number of possible even without a calculator. Now you can easily cope with any stitch multiplication table 9.

For example, if we need to multiply 9 by 9, proceed as follows:

9*9 = 90-9 = 81

If needed multiply 9 by 3:

9*3 = 3*9 = 30-3 = 27

In all these transformations have no miracles, we just use the normal math.

9 multiply

My personal opinion is: better to use the thinking abilities of their brain than clog the brain remembering any nonsense. Even if this nonsense forced to learn mathematics.

Stupid triangle top view

Maybe it's someone not very pleasant, but stupid triangles does not happen, there are only stupid people. But wits on the Internet are often looking for a stupid triangle, top view which they would very much like to see. Show.

Stupid triangle top view

First, let's talk about the name of this type of triangles. Call similar triangles stupid - a sign of illiteracy. As they say now, "politically correct" will call such triangles "obtuse". All the triangles which have one angle greater than 90 degrees, are obtuse triangle. It's not a shame, is not a defect, just that physique is obtuse triangles. However, each obtuse triangle always has a couple of sharp corners. So, just in case.

Now let's talk about the kinds. No triangles on the kinds and types of top side of the window. Obtuse triangle top looks exactly the same as the bottom. But the side view is not pleasing to the eye - is just a regular interval. From the window you are unlikely to consider anything, but in a notebook at a neighbor's party this triangle you can see from many angles. In this case the obtuse triangle will look quite different, what it looks like from above. Describe such a wonderful transformation is possible by means of projective geometry, descriptive geometry, trigonometry, or poems. Who like more.

How not to change the signs inside the brackets?

All of us are taught to change signs when opening or closing brackets of the expression in brackets, if before the brackets minus sign. Let's look at this tedious process for half-dead examples.

11-(2+5-4) = 11-3 = 8

Before the expression in brackets is a minus sign, it means that the disclosure of the brackets need to change all the signs to the contrary all the numbers that are inside the parentheses. Consider the same example, but without the brackets.

11-(2+5-4) = 11-2-5+4 = 9-5+4 = 4+4 = 8

Now let's try to take a part of the expression in parentheses.

1+2+3+4 = 10

Naturally, you ask: "Where is the minus sign?!" Do not worry, now appear.

1+2-(-3-4) = 3-(-7) = 3+7 = 10

I put in front of the brackets and the minus sign to change the sign before the numbers inside the parentheses. In disclosing the brackets again I changed the sign, because before my braces a minus sign. In the end, the result remained unchanged.

Now a more complex example.

17-6+9 = 20
17-(6-9) = 17-(-3) = 17+3 = 20

As you can see, continuous headache turns out, when suddenly appears before the brackets the minus sign. How not to change the signs inside the brackets? Very simply - no need to put a minus in front of the brackets. Look how it's done.

17-6+9 = 17+(-6+9) = 17+(3) = 17+3 = 20

Now consider the last two examples under the microscope. In the first case I put the first parenthesis after the minus sign. I like a knife cut a negative number into two parts - a minus sign and a positive number. The minus sign was in front of the bracket, and a positive number - within the brackets. Look.

17-(6........

In fact, we are in parentheses conclude a positive number, which until then was negative. Changing the sign of the first number in parentheses passed on complete autopilot without our intervention. A sort of a machine circumcision minus sign in numbers. But with the other numbers that fall into these brackets are already having problems. Signs they need to be changed manually.

In the second case I put the opening bracket before the minus sign. In fact, I put in brackets a negative number with a minus sign. Here's how it looks initially.

17(-6.........

Now, between numbers 17 and bracket there is no sign that mathematics implies multiplication. But I do not have anything to multiply. To answer in the solution remained the same example, I set the bracket additional "plus" sign.

17+(-6.........

Now everything is correctly written. Before the parentheses appears the plus sign and signs before the numbers inside the brackets intact. No mathematical crime I did not commit, just competently get rid of extra action on the replacement characters inside the brackets. Why mathematics always do? They had not taught this. If it is not in the curriculum, and then teach you that no one will. Math enough to know, you need to also know how to use her.