1.4 Divisors

A divisor is a number that when divided by itself gives the remainder as zero.

How do we find a number's divisor? That is where our own robot, Emily, and Maxima come in.

Type divisors(24) in Emily to find all of a given number's divisors. You will get 1, 2, 3, 4, 6, 8, 12, 24.

Maxima is a free computer algebra system that can be installed in Windows, Mac OS, and Linux. It is very powerful, fast, and easy to use, ideal for students, teachers, and researchers alike.

You just type in an expression and press "Shift" and "Enter", e.g. the query divisor(24); gives back the divisor of 24.


1.5 Primes

A prime number is one that has exactly two divisors: the number 1 and the number itself.

Type the query isPrime(21) to determine whether a number is prime or not and press ENTER.
primeList(20) =2 3 5 7 11 13 17 19 returns a list of prime numbers [2..20].
mathQuiz(5, 4) makes Emily ask you five questions about primes.

The command primep() is useful for testing whether or not an integer is prime or not, e.g. 7 is a prime number, but 36 is not. Wolfram Alpha uses natural language processing to understand queries such as: is 7 a prime number?

primes<=70; --WolframAlpha
pimep(36); --Maxima


1.6 Prime factorization

Prime factorization is the discovery of which prime numbers you need to multiply together to get a certain number.

Type factor(180)=22 * 32 * 5 to calculate a given number's prime factorization.

Observe that factor(420); gives back the prime factorization of 420 in Maxima and WolframAlpha, the only difference is the use of a semicolon (;) in Maxima.

factor(420) --WolframAlpha
factor(420); --Maxima


1.7 Greatest common divisor (gcd) and least common multiple (lcm)

The greatest common divisor (gcd) of two numbers (420, 54) is the largest product of the primes shared by the two numbers(2*3); the least common multiple (lcm) is the smallest number that is a multiple of both numbers (2*2*3*3*3*5*7).

Typing gcd(96,90) =6 into Emily returns the greatest common divisor of two given numbers. Additionally, lcm(96,90) =1440 returns the least common multiple of two given numbers.

lcm(420,54) --WolframAlpha
gcd(420,54); --Maxima


1.8 Powers and roots

The power of a number is the amount of times that number is multiplied by itself. For example, 7 to the power of two is 49 (7 * 7), 4 to the power of three is 64. To raise a number n to a power p in Maxima, simply type n^p.

The square root of a number is the one that, when multiplied by itself, is equal to the original number. For example, the square root of 49 is 7, the query in Maxima is sqrt(49);

x^y gives "x" to the power "y" and sqrt(z), the square root of z. For example, 4 ^ 2 =42= 16, and sqrt(9) =√9 = 3

sqrt(64) --WolframAlpha
4³; --Maxima
inrt(64, 3);


1.9 Roman numerals

Roman numerals is the numeric system used in ancient Rome. They are written as combinations of seven letters: I, V, X, L, C, D, and M. Each letter represents a number: I is 1, V is 5, X is 10, and so on. For instance, the numbers 1 to 15 can be expressed in Roman numerals as follows: I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XV.

numToRoman(1254) =MCCLIV converts a number into Roman numerals. mathQuiz(5, 8) tests your knowledge of Roman numerals.

1.10 Others number systems: binary, octal, and hexadecimal.

A number system is a "writing system for expressing numbers" (Wikipedia). The most common one used today is called the decimal system. It has ten symbols (0-9) and it has been so widely adopted throughout the world that you may not know that other number systems exist.

The binary system has a base of two, and uses only two symbols: 0 and 1. Therefore, when you count in binary, it goes like 0 (0), 1(1), 10(2), 11(3), 100(4), 101(5), 110(6), 111(7)..., so each place value is a power of 2. 10102 = 1*23 + 1*21 = 10. It is widely used in digital computers.

The octal numeral system is the base-8 number system. It has eight symbols starting from 0 to 7. This system is also a positional number system, which means that the value of each digit depends on the position of the digit in the numeral. For example: 2758 = 2*82 + 7*81+ 5*80 = 189.

The hexadecimal system has a base of sixteen, and it uses sixteen different symbols: 0, 1, 2, ... 9, A(10), B(11), C(12), D(13), E(14), F(15). In this system, each weight equals 16 raised to the power of its position. For example: 1F216 = 1 * 162 + 15*161 + 2 * 160 = 498

Elisa can handle conversions between different number systems: dec2Bin(54) =1101102 , decimal to binary; dec2Oct(14) = 168 , decimal to octal; dec2Hex(45) = 2D16 , decimal to hexadecimal; bin2Dec(11011) = 27, binary to decimal; oct2Dec(27) = 23, octal to decimal; hex2Dec(AF) = 175, hexadecimal to decimal.

2. Integers

Integers are whole numbers(1, 2, 3, 4,...), negative whole numbers(-1, -2, -3, -4,...), and zero(0), but there are still no fractions allowed. They are used for expressing temperatures that are below 0ºC, for measuring altitude below sea level, and when one withdraws money from a bank account. Google, WolframAlpha, and Maxima can handle integers.

"If a person is engaged in trade, his capital will be increased by his gains, and diminished by his losses. Increase in temperature is measured by the number of de- grees the mercury rises in a thermometer, and decrease in temperature by the number of degrees the mercury falls. In considering any quantity whatever, a quantity that in- creases the quantity considered is called a positive quantity; and a quantity that decreases the quantity considered is called a negative quantity." (The First Steps in Algebra, G. A. (George Albert) Wentworth).

The absolute value of a number is its distance from zero on the number line: |-4| = 4, |7| = 7. The opposite of a number is the number on the opposite side of the number line: -(-7)=7.

Operations with integers are a little tricky. Here is what we do when we need to add a negative and a positive number (-8 + 7). Use the sign of the larger number and subtract the smaller absolute value from the larger absolute value: -1 (8 -7 = 1).

Remember: (R1) The product of a positive integer and a negative integer is always a negative integer: 4 * -3 = 12, -5 * 3 = 15. The product of two negative integers (-3 * -3 = 9, -4 * -5 = 20) or two positive integers (2 * 4 = 8, 4 * 7 = 28) is always a positive integer. The same rules apply to division: R1 (8 : 4 = 2, -9 : -3 = 3), R2 (8 : -4 = -2, -4 : 2 = -2).

Emily can handle integers, too: -3 * 5, 2 + -3, (4 + -7) * -2, etc. abs(-7) =7 computes the absolute value of a given number. mathQuiz(10, 5) returns a quiz with ten questions about integers with numbers 1-10; mathQuiz(10, 6) challenges you with questions about integers from 1 to 100; mathQuiz(10, 7) asks you questions about integers with numbers 1-1000.

You can see the representation of an integer on a Number Line with WolframAlpha. The absolute value is the distance of that number from zero. The absolute value of a positive number is that very number, abs(9)=9. The absolute value of zero is zero, abs(0)=0; and the absolute value of a negative number is its opposite, abs(-7)=7

3+(-5) --WolframAlpha
4-5; --Maxima