Category: College Mathematics

How to Change Number Bases Part 1

Number Theory

I have already discussed clock arithmetic,  modulo division, and number bases. We further our discussion in this post by learning how to change numbers from one base to another.

The number system that we are using everyday is called the decimal number system or the base 10 number system (deci means 10). It is believed that this system was developed because we have 10 fingers.

In the base 10 system, the digits are composed of  0 up to 9. Adding 1 to 9, the largest digit in this system, will give us 10. That is, we replace 9 in the ones place with 0, and add 1 to the tens place which is the next larger place value.

Another way to write a number in base 10 is by multiplying its digits by powers of 10 and adding them. For example, the number 2578 can be rewritten in expanded form as

2(10^3) + 5(10^2) + 7(10^1) + 8(10^0).  » Read more

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Banach-Tarski Paradox and A Great Discussion About Infinity

College Mathematics

This video is a rich discussion about about infinity. It starts from the famous infinite chocolate problem, countable infinity, uncountable infinity, Cantor’s diagonalization system, Hilbert’s hotel, and the Banach-Tarski Paradox. Although the last part is a bit too much for the level of this blog, the explanations before it of the concepts before it are extremely clear and helpful.

I had several discussions about infinity in this blog particularly the content of Counting the Real Numbers, The Grand Hotel Paradox, and One to One Correspondence. You might want to check them out.

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Understanding If-Then Statements

College Mathematics, High School Mathematics

If I get my bonus, then I will buy a car.
If I become 30, then I will get married.
If x \geq 3, then x^2 \geq 9.

The statements above are examples of an if-then statement. An if-then statement is also called a conditional statement. A conditional statement is of the form if p then q where p and q are statements and in which q follows from p. As shown above, conditional statements can be seen in real life or in mathematics.

In a conditional statement, the statement following if is called the antecedent (or hypothesis), while the statement following then is called the consequent (or conclusion).

Conditional statements have their truth values just like, and, or and not. To examine their truth values, let us consider the third conditional statement above. In this statement, we can assign p as x \geq 3  and q as x^2 \geq 9.  » Read more

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