Math Nuggets

A divisor of a prime power sum will also divide infinite other prime power sums.

2022-12-10T15:15:00.008+11:00

For the past two years I've been looking at sums of primes raised to various powers and seeing what can divide them. In particular, if you sum the first n primes each raised to a power k, when is the sum divisible by n? For example, the sum of the squares of the first 19 primes is divisible by 19: 2**2 + 3**2 + 5**2 + ... + 67**2 = 19*1314 so 19 is a solution for the second power (i.e. k = 2

Greatest Common Divisor (GCD), Least Common Multiple (LCM)

2009-05-16T16:12:00.003+10:00

This calculator takes two numbers and finds the greatest common divisor and the least common multiple. var fnhash1 = { mygcdlcm : {fn: function (v) { v[0] = Math.abs(parseInt(v[0])); v[1] = Math.abs(parseInt(v[1])); a = v[0]; b = v[1]; while (b!=0) { t = b; b =

Prime Factors Calculator

2009-05-01T22:51:00.002+10:00

I made this javascript prime factors calculator and have found it addictive. I just keep factorizing numbers. Enter the number to factorize in the left box and click on the 'factorize' button. The prime factors are shown in the right box. function factor(x) { if (x >= 9007199254740992) return 'too big for me'; if (x==1) return ''; m = Math.sqrt(x) + 0.5 for (i=2;i');

The Numbers go Social Networking

2009-04-22T09:57:00.003+10:00

Names and photos are clickable. Pythagoras is thinking that a2 + b2 = c2 for right angled triangles. 12 hours ago · Comment · Like Natural at 10am on 14 April Likes this! Rational at a ¼ past 10 on 14 April like 3,4,5 & 20,21,29

When are addition, subtraction, multiplication, division and exponentiation allowed?

2009-04-18T14:47:00.014+10:00

Now that we've had a look at several groups of numbers let's bring together what operations are allowed for each one: +-×÷ab natural numbersyesonly larger number minus smaller or equal numberyesonly if it divides evenly; can't divide by zeroyes integersyesyesyesonly if it divides evenly; can't divide by zeroonly positive and zero powers rational numbersyesyesyescan't divide by

Quaternions

2009-04-18T10:10:00.002+10:00

The quaternions are an extension of the complex numbers. Instead of having just one square root of minus one: i, why not have 3: i, j and k? We will need a way of multiplying them together. It turns out that the following works: ij = k = -ji jk = i = -kj ki = j = -ki ijk = -1 The quaternions are not commutative, which is quite strange. The order in which you multiply them matters! If you

Complex Numbers

2009-04-18T10:08:00.001+10:00

The complex numbers are the combination of the real numbers with the imaginary numbers. They are written as the real part plus the imaginary part. For example: 2 + 3i The imaginary number i is defined as the square root of minus one, so i2 = -1. Multiplication of complex numbers follows the same rules as the real numbers, you just have to keep track of each part of the multiplication: (2 +

Imaginary Numbers

2009-04-18T10:06:00.002+10:00

The imaginary numbers are needed to fill in the last gap in exponentiation. The real numbers do not allow some negative numbers to be raised to fractional powers. For example they do not allow (-1)(1/2) = √(-1). To get around this problem we can just make up an answer. We'll call this answer i. It can't be a real number, so it is outside the real numbers. Using i allows us to find roots

Transcendental Numbers

2009-04-18T10:04:00.018+10:00

The transcendental numbers are a part of the irrational numbers. Transcendental numbers are numbers that do not solve any polynomial equation that has rational coefficients. π and e are transcendental numbers so they don't solve any equations like: 4x - 7 =0 5x2 - 3x + 4 = 0 (4/5)x7 - 1.24x3 - 8 = 0 Strangely, it is not known whether π + e or π × e is transcendental,

Real Numbers

2009-04-18T09:54:00.005+10:00

If we combine the rational numbers with the irrational numbers we get the real numbers. In the real numbers we find positive numbers, negative numbers, integers, fractions, square roots, cube roots, fractional roots, π, e, zero... Like the rational numbers, we can add, subtract, multiply and divide any two numbers (except dividing by zero). But with the real numbers we can also raise any

Irrational Numbers

2009-04-17T20:27:00.010+10:00

Just as rational numbers were made up by ratios, the irrational numbers cannot be expressed as a ratio. For example the square root of 2 does not equal one integer divided by another. You can get closer and closer with fractions, but you will never exactly equal √2. This also means that the decimal expansion of an irrational number goes on forever. Square roots, cube roots and other

Rational Numbers

2009-04-17T20:12:00.007+10:00

We grouped the negative numbers with the natural numbers to get the integers. Now we increase our set of numbers further to get the rational numbers. The rational numbers include all of the fractions made by dividing one integer by another, except that you can't divide by zero. So positive and negative fractions, fractions smaller than one, fractions larger than one, and all the integers (you

Integers

2009-04-16T22:48:00.004+10:00

"Dad, can you buy me those butterfly wings?" "They are seven dollars. How about you use some of your own money?" "But I don't have my money with me." "If I buy the wings, you can owe me the money and give it to me when we get back home." Shortly afterwards my daughter has the butterfly wings and -$7 in her pocket and is thus introduced to negative numbers, and spending on credit. The first

Natural Numbers

2009-04-15T16:17:00.003+10:00

"What do you get when you take away 7 from 3?", I ask my daughter. "You can't do that. That's silly", she replies. "What about if I divide 5 in half?" "You get 2 in one group and 3 in the other." Such in the world of a child, and such is the world of the natural numbers. In this world you start at zero and count 1, 2, 3 and so on. That's all the numbers you've got. There's no problem with

Associative, but not commutative

2009-04-13T18:19:00.006+10:00

Every operation mentioned in the previous post that is associative is also commutative, and everything mentioned that is not associative is not commutative. associativecommutative additionyesyes mutiplicationyesyes subtractionnono divisionnono exponentiationnono So is there anything that is associative but not commutative? Let's look at rotation. Can we show that rotation is not commutative?

Would you eat chocolate milk powder? (associativity)

2009-04-12T09:00:00.012+10:00

"Yuk! Why are you eating Milo* straight?", I ask my daughter. "Isn't it better mixed in milk?" "I've already drunk the milk", she replies, "so now I'm eating the Milo." Is drinking milk and then eating Milo, the same as first mixing the Milo in the milk and then drinking the result? Maybe your stomach doesn't notice, but it certainly tastes different. To write this another way, are these things

Put on your shoes and socks: commutation

2009-04-11T09:23:00.012+10:00

"Put on your shoes and socks - but not in that order", I say to my daughter, who sighs at my frequently repeated joke. The order you put your clothes on matters. Putting on your shoes then your socks is quite different from putting on your socks first and then your shoes. On the other hand, it doesn't matter which sock you put on first, your left or your right. In mathematics, when the order

Continued fractions

2009-04-09T17:45:00.011+10:00

Any whole number or simple fraction can be written as a continued fraction. Continued fractions appear a little strange and look like this: 1 a0 + ---------------------- 1 a1 + --------------- 1 a2 + -------- a3 + ... To make it easier to write down, the above fraction is written as: [a0 ; a1, a2, a3 ...] For example, the

Pi by fractions

2009-04-08T20:36:00.010+10:00

The circumference of a circle divided by its diameter is always the same number, no matter the size of the circle. This number is called π (pi) and has the value 3.1415926535... π is an irrational number, which means that it cannot be expressed as a simple fraction such as 1/5 or 3/4. However, you can get pretty close. A fraction that is often used is 22/7. This is not all that good: 22/7 =

More Pythagorean Triangles

2009-04-08T16:09:00.009+10:00

Here's another rule for generating Pythagorean trianges, Last time each triangle began with an odd number. This time we will use even numbers: 4, 3, 5 (16 + 9 = 25) 6, 8, 10 (36 + 64 = 100) 8, 15, 17 (64 + 225 = 289) You generate first number of the next line by writing down the next even number. To find the second and third numbers, add the first number to the second number in the previous

Generate Pythagoras Triangles

2009-04-07T21:34:00.007+10:00

Triangles containing a right angle are called Pythagorean triangles. Their side lengths have a simple relationship. The sum of the squares of the shorter side lengths equals the square of the length of the longer side. A simple example is a triangle with side lengths: 3, 4 and 5 since 32 + 42 = 9 + 16 = 25 = 52 Some other examples are: 5, 12, 13 (25 + 144 = 169) 6, 8, 10 (36 + 64 = 100) 7,

Cube Sum is a Square

2009-04-05T20:13:00.015+10:00

Pick a whole number. If you add up the cubes of all the whole numbers from 1 to your number, you will get the same as if you add up all the numbers from 1 to your number and then square the result. For example choose the number 3: Add up the cubes of all the numbers from 1 to 3: 13 + 23 + 33 = 1 + 8 + 27 = 36 Now, add up all the numbers from 1 to your number 1 + 2 + 3 = 6 and then square

6174

2009-04-05T17:28:00.007+10:00

Take a four digit number. Put the digits in order from smallest to biggest to make number A. Then put them in order from biggest to smallest to make number B. Calculate B minus A to get the answer. Repeat this process with the answer, and keep repeating. For example start with 7119. Putting the digits in order from smallest to largest: 1179 Now from largest to smallest: 9711 Subtract: 9711 -

Halve and Triple

2009-04-05T12:02:00.007+10:00

Think of a whole number. If it's even, halve it. If it's odd, triple it and add one. Then repeat this procedure with the answer. Continue until you reach 1. Did you get to 1 quickly? Or did it take a long time, going through many numbers? It depends which number you start with. 8 is very quick, taking only three turns: 8, 4, 2, 1. But 9 takes 19 turns! (9, 28, 14, 7, 22, 11, 34, 17, 52, 26,