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Showing posts with label number. Show all posts
Showing posts with label number. Show all posts

1.01.2022

Signs greater than and less than

How to remember signs greater than and less than? A very long time ago, when I was little and went to school, the teacher taught us this. Even as an adult, I have always used this method.

We always do more with our right hand than with our left. If we bend our right arm at the elbow and raise it above our head, we get the "GREATER THAN" sign. We do less with our left hand. A bent and raised left hand will give us the "LESS THAN" sign. The elbow of the right hand points to the direction of the "GREATER THAN" sign. This is how it looks in the picture.

 Signs greater than and less than

Indeed, the arm bent at the elbow is very similar to the "greater than" and "less than" symbols. You don't even have to raise your hand. And this cheat sheet will always be with you. If mathematicians don't cut your hands off.

If you are left-handed and do more with your left hand than with your right? How then to be? You have two options. Or you think like everyone else and you won't have any problems with math. Or you write your "left-handed math" and learn a lot of interesting things.

10.30.2021

2 to the power zero five

How much is 2 to the power zero five? A trick question, like everything mathematicians come up with for us. In short, transformations look like this.

 2 to the power zero five

Now let's take a closer look at the logic of mathematicians. Raising a number to an integer power is the multiplication of these numbers. Raising a number to a fractional power is extracting the root of a number, with the number in the denominator showing which root to find. But the fraction itself can be written both in the form of a decimal fraction, and in the form of a regular fraction. We put all this in a pot, mix thoroughly and the mathematical puzzle for the textbook is ready.

You need to solve this puzzle in the following order. We convert the decimal fraction zero five into a regular fraction and get one second. Now we can write the number 2 with a fractional exponent as the square root of two and calculate its value.

To consolidate knowledge, consider two more simple examples.

 4 to the power zero five

4 to the power zero five is the same as 4 to the one-second power. Four to the power of one second is the square root of 4. Taking the square root of four gives two.

 25 to the power zero five

After considering the two previous examples, it will be very easy to calculate 25 to the power zero five. As real shamans, we turn zero five into one second, extract the square root of 25 and get the desired result - the root of twenty-five is five.

Why all these dances of shamans with tambourines? There are two sides to this math medal. Front side - mathematicians teach us to use mathematics. The other side of the coin is that if mathematicians simply and clearly express their thoughts in the language of mathematics, then they will not seem so smart to us, and we ourselves will not look such fools.

7.25.2021

Cube of 101

Today we will look at an example of how you can find one hundred and one cubed using the abbreviated multiplication formulas. In other words, how to raise cube of 101 using the abbreviated multiplication formula.

The following cry for help was heard in the comments:

Help please, problem (example) from the physics and mathematics school (Russia), grade 8:

101^3

I don't understand how to solve this with abbreviated multiplication formulas.

It is very easy to cube the number 101 - you need to type the number 101 on a calculator and multiply twice by the same number 101.

101^3=101*101*101=1030301

If you don't have a calculator at hand (you never know, the phone has just been stolen), then you can calculate on a piece of paper in a column (the picture will be at the end, like checking both the calculator and the formula).

Cube of 101 We apply the cube of sum formula.

The problem statement says that you need to find the cube of 101 using the abbreviated multiplication formula. According to math teachers, everyone should know these formulas. Naive. Where to find this formula?

You can search the Internet for a cube of sum. Google to help you. You can find these formulas in a reference book on mathematics, in a textbook on mathematics, you can ask a classmate who knows the formulas for abbreviated multiplication by heart. The formula for the abbreviated multiplication we need is called cube of sum. You will see it below.

And now the answer to the most tricky question: how to get the sum of numbers from one number? It is necessary to expand this number into terms. From the point of view of mathematics, the number of terms can be any, but ... I found the abbreviated multiplication formulas for raising a sum to a cube only for two and three terms. The formula for the cube of the sum of three terms is very complicated, I wish you never to face such a thing. But the cube of the sum of the two terms looks nice. The basic principle of expansion into terms for applying abbreviated multiplication formulas is that numbers can be easily multiplied in your head without using a calculator. For the number 101, the best option would be 101 = 100 + 1. 100 and 1 numbers is easy to multiply without a calculator. Let's see what we get.

 Cube of 101

I don't know about you, but I couldn't do without a piece of paper. Yes, I wrote everything down in a line, not a column, but nevertheless. And in conclusion, let's check our solution by multiplying in a column on a piece of paper.

 Cube of 101 in a column on a piece of paper

Well, something like this. Why do you need all this? So that you know that you can multiply not only on a calculator or in a column, but sometimes you can effectively use abbreviated multiplication formulas. If, of course, you know them.

12.20.2020

Plus on minus what gives?

Positive and negative numbers were invented by mathematicians. The rules for multiplying and dividing positive and negative numbers were also invented by mathematicians. We need to learn these rules in order to tell mathematicians what they want to hear from us.

It's easy to remember the rules for multiplying or dividing positive and negative numbers. If two numbers have different signs, the result will always be a minus sign. If two numbers have the same sign, the result will always be a plus. Let's consider all possible options.

What gives plus for minus? When multiplying and dividing, plus by minus gives a minus.

 What gives plus for minus?

What gives a minus on plus? When multiplying and dividing, we also get a minus sign as a result.

 What gives a minus on plus?

As you can see, all the options for multiplying or dividing positive and negative numbers have been exhausted, but the plus sign has not appeared.

What gives a minus on a minus? There will always be a plus if we do multiplication or division.

 What gives a minus on a minus?

What gives plus for plus? It's quite simple here. Multiplying or dividing plus by plus always gives plus.

 What gives plus for plus?

If everything is clear with the multiplication and division of two pluses (the result is the same plus), then with two minuses, nothing is clear.

Why does a minus and a minus make a plus?

I can assure you that, intuitively, mathematicians have correctly solved the problem of multiplying and dividing the pros and cons. They wrote down the rules in textbooks without giving us any reasons. For the correct answer to the question, we need to figure out what the plus and minus signs mean in mathematics.

One mathematics teacher told us in the classroom: "Mathematics is an exact science, if you lie twice, you get the truth". This statement was very useful to me. Once I was solving a difficult problem with a long solution. I knew exactly what the result should be. But the result was different. I have been looking for an error in the calculations for a long time, but I could not find it. Then, a few steps before the final result, I changed one number so that the result was correct. I lied twice in the calculations and got the correct result. This is very similar to the minus on minus equals plus rule, isn't it?

2.10.2017

Number to the power minus one

What to do if number to the power minus one? Write down this number in a fraction denominator. Here several examples with numbers to the power minus one.

 Number to the power minus one
Negative exponents of numbers mean that these numbers are in a fraction denominator.

 Negative exponents

1.07.2015

Happy New Year!

 Happy New Year!
Happy New Year to you all! Do not look at the numbers in 2015, they may look much prettier. It all depends on what we want to see them.

 2015 year
The number 2015 in the binary system looks much nicer and has an almost perfect symmetry. But it will look like a calendar for 2015 in the binary system.

 Calendar for 2015
And New Year's letter to Santa Claus. To the Russian Santa Claus (Ded Moroz) from Ukrainian girl whose father was killed in the Russian-Ukrainian war. No comment.
 New Year's letter to Santa Claus from Ukrainian girl

"Good afternoon, Santa Claus!
I have two big dreams.
My first dream is to end the war and did not die, our Pope!
Second, I want the laptop to communicate with children who also lost their dads, and who also want peace.
Thank you for understanding.
Please help me in my dreams.
From Levchuk Marija"

Here's how Putin congratulates Ukrainian people on the New Year.

 Happy New Year from Putin

And here is how the humanitarian assistance from Putin in Ukraine.

 Humanitarian assistance from Putin in Ukraine

12.18.2014

How to find the number of

Hello. Please help me to solve the problem.
Subject: decimals.
The sum of three numbers is 10 whole 4/10. The second number is 3 times smaller than the first, and at one whole 1/10 more than the third. Find these numbers.

Carefully read the conditions and form the equation. "The second number is 3 times smaller than the first" - this means that the first number is three times langer than the second. If the second number is denoted by X, then the first number is three X.

"The second number is (((3 times smaller than the first, and))) at one whole 1/10 more than the third" - means the third number is equal to X minus one whole 1/10.

Now write the sum of three numbers equate it to 10.4 and obtain the equation.

3x + x + x - 1.1 = 10.4
5x = 10.4 + 1.1
5x = 11.5
x = 11.5 / 5
x = 2.3

The second number is equal to 2.3
The first number is equal to 2.3 * 3 = 6.9
The third number is 2.3-1.1 = 1.2

Check the amount of received numbers

6.9 + 2.3 + 1.2 = 10.4