What is a prime number? How can you find prime numbers?
A prime number is a positive integer that has exactly two positive integer factors, 1 and itself. For example, if we list the factors of 28, we have 1, 2, 4, 7, 14, and 28. That's six factors. If we list the factors of 29, we only have 1 and 29. That's two factors. So we say that 29 is a prime number, but 28 isn't.
Another way of saying this is that a prime number is a positive integer that is not the product of two smaller positive integers.
Note that the definition of a prime number doesn't allow 1 to be a prime number: 1 only has one factor, namely 1. Prime numbers have exactly two factors, not "at most two" or anything like that. When a number has more than two factors it is called a composite number.
Here are the first few prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.
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What's the 'Sieve of Eratosthenes'?
The Sieve of Eratosthenes
Eratosthenes (275-194 B.C., Greece) devised a 'sieve' to discover prime numbers. A sieve is like a strainer that you use to drain spaghetti when it is done cooking. The water drains out, leaving your spaghetti behind. Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind.
To use the sieve of Eratosthenes to find the prime numbers up to 100, make a chart of the first one hundred positive integers (1-100):
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
Cross out 1, because it is not prime.
Circle 2, because it is the smallest positive even prime. Now cross out every multiple of 2; in other words, cross out every second number.
Circle 3, the next prime. Then cross out all of the multiples of 3; in other words, every third number. Some, like 6, may have already been crossed out because they are multiples of 2.
Circle the next open number, 5. Now cross out all of the multiples of 5, or every 5th number.
Continue doing this until all the numbers through 100 have either been circled or crossed out. You have just circled all the prime numbers from 1 to 100!
The First 1,000 Primes
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67 71
73 79 83 89 97 101 103 107 109 113
127 131 137 139 149 151 157 163 167 173
179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281
283 293 307 311 313 317 331 337 347 349
353 359 367 373 379 383 389 397 401 409
419 421 431 433 439 443 449 457 461 463
467 479 487 491 499 503 509 521 523 541
547 557 563 569 571 577 587 593 599 601
607 613 617 619 631 641 643 647 653 659
661 673 677 683 691 701 709 719 727 733
739 743 751 757 761 769 773 787 797 809
811 821 823 827 829 839 853 857 859 863
877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997 1009 1013
1019 1021 1031 1033 1039 1049 1051 1061 1063 1069
1087 1091 1093 1097 1103 1109 1117 1123 1129 1151
1153 1163 1171 1181 1187 1193 1201 1213 1217 1223
1229 1231 1237 1249 1259 1277 1279 1283 1289 1291
1297 1301 1303 1307 1319 1321 1327 1361 1367 1373
1381 1399 1409 1423 1427 1429 1433 1439 1447 1451
1453 1459 1471 1481 1483 1487 1489 1493 1499 1511
1523 1531 1543 1549 1553 1559 1567 1571 1579 1583
1597 1601 1607 1609 1613 1619 1621 1627 1637 1657
1663 1667 1669 1693 1697 1699 1709 1721 1723 1733
1741 1747 1753 1759 1777 1783 1787 1789 1801 1811
1823 1831 1847 1861 1867 1871 1873 1877 1879 1889
1901 1907 1913 1931 1933 1949 1951 1973 1979 1987
1993 1997 1999 2003 2011 2017 2027 2029 2039 2053
2063 2069 2081 2083 2087 2089 2099 2111 2113 2129
2131 2137 2141 2143 2153 2161 2179 2203 2207 2213
2221 2237 2239 2243 2251 2267 2269 2273 2281 2287
2293 2297 2309 2311 2333 2339 2341 2347 2351 2357
2371 2377 2381 2383 2389 2393 2399 2411 2417 2423
2437 2441 2447 2459 2467 2473 2477 2503 2521 2531
2539 2543 2549 2551 2557 2579 2591 2593 2609 2617
2621 2633 2647 2657 2659 2663 2671 2677 2683 2687
2689 2693 2699 2707 2711 2713 2719 2729 2731 2741
2749 2753 2767 2777 2789 2791 2797 2801 2803 2819
2833 2837 2843 2851 2857 2861 2879 2887 2897 2903
2909 2917 2927 2939 2953 2957 2963 2969 2971 2999
3001 3011 3019 3023 3037 3041 3049 3061 3067 3079
3083 3089 3109 3119 3121 3137 3163 3167 3169 3181
3187 3191 3203 3209 3217 3221 3229 3251 3253 3257
3259 3271 3299 3301 3307 3313 3319 3323 3329 3331
3343 3347 3359 3361 3371 3373 3389 3391 3407 3413
3433 3449 3457 3461 3463 3467 3469 3491 3499 3511
3517 3527 3529 3533 3539 3541 3547 3557 3559 3571
3581 3583 3593 3607 3613 3617 3623 3631 3637 3643
3659 3671 3673 3677 3691 3697 3701 3709 3719 3727
3733 3739 3761 3767 3769 3779 3793 3797 3803 3821
3823 3833 3847 3851 3853 3863 3877 3881 3889 3907
3911 3917 3919 3923 3929 3931 3943 3947 3967 3989
4001 4003 4007 4013 4019 4021 4027 4049 4051 4057
4073 4079 4091 4093 4099 4111 4127 4129 4133 4139
4153 4157 4159 4177 4201 4211 4217 4219 4229 4231
4241 4243 4253 4259 4261 4271 4273 4283 4289 4297
4327 4337 4339 4349 4357 4363 4373 4391 4397 4409
4421 4423 4441 4447 4451 4457 4463 4481 4483 4493
4507 4513 4517 4519 4523 4547 4549 4561 4567 4583
4591 4597 4603 4621 4637 4639 4643 4649 4651 4657
4663 4673 4679 4691 4703 4721 4723 4729 4733 4751
4759 4783 4787 4789 4793 4799 4801 4813 4817 4831
4861 4871 4877 4889 4903 4909 4919 4931 4933 4937
4943 4951 4957 4967 4969 4973 4987 4993 4999 5003
5009 5011 5021 5023 5039 5051 5059 5077 5081 5087
5099 5101 5107 5113 5119 5147 5153 5167 5171 5179
5189 5197 5209 5227 5231 5233 5237 5261 5273 5279
5281 5297 5303 5309 5323 5333 5347 5351 5381 5387
5393 5399 5407 5413 5417 5419 5431 5437 5441 5443
5449 5471 5477 5479 5483 5501 5503 5507 5519 5521
5527 5531 5557 5563 5569 5573 5581 5591 5623 5639
5641 5647 5651 5653 5657 5659 5669 5683 5689 5693
5701 5711 5717 5737 5741 5743 5749 5779 5783 5791
5801 5807 5813 5821 5827 5839 5843 5849 5851 5857
5861 5867 5869 5879 5881 5897 5903 5923 5927 5939
5953 5981 5987 6007 6011 6029 6037 6043 6047 6053
6067 6073 6079 6089 6091 6101 6113 6121 6131 6133
6143 6151 6163 6173 6197 6199 6203 6211 6217 6221
6229 6247 6257 6263 6269 6271 6277 6287 6299 6301
6311 6317 6323 6329 6337 6343 6353 6359 6361 6367
6373 6379 6389 6397 6421 6427 6449 6451 6469 6473
6481 6491 6521 6529 6547 6551 6553 6563 6569 6571
6577 6581 6599 6607 6619 6637 6653 6659 6661 6673
6679 6689 6691 6701 6703 6709 6719 6733 6737 6761
6763 6779 6781 6791 6793 6803 6823 6827 6829 6833
6841 6857 6863 6869 6871 6883 6899 6907 6911 6917
6947 6949 6959 6961 6967 6971 6977 6983 6991 6997
7001 7013 7019 7027 7039 7043 7057 7069 7079 7103
7109 7121 7127 7129 7151 7159 7177 7187 7193 7207
7211 7213 7219 7229 7237 7243 7247 7253 7283 7297
7307 7309 7321 7331 7333 7349 7351 7369 7393 7411
7417 7433 7451 7457 7459 7477 7481 7487 7489 7499
7507 7517 7523 7529 7537 7541 7547 7549 7559 7561
7573 7577 7583 7589 7591 7603 7607 7621 7639 7643
7649 7669 7673 7681 7687 7691 7699 7703 7717 7723
7727 7741 7753 7757 7759 7789 7793 7817 7823 7829
7841 7853 7867 7873 7877 7879 7883 7901 7907 7919
3-D Shapes
THREE-DIMENSIONAL SHAPES
Now we've gone and done it. 3-D. That term stands for three-dimensional. It means an object that has depth. It's an object that you can hold. You are a three-dimensional object. Your computer is a three-dimensional object. Your balls in the toy chest are three-dimensional objects. Everything in the world is three-dimensional. 3-D objects with corners are called polyhedrons in math. It's a word related to polygon (2-D objects). Let's look at a few basic shapes...
SPHERES AND SPHEROIDS
You know that ball you use on the playground? That ball… That perfectly round ball… Is a sphere. Spheres are smooth surfaces with no edges. How do you make a sphere? Take a circle and spin it around its centerline. It is just that simple. Spheroids are round objects that weren't made with perfect circles. A great example of a spheroid is an egg.
PYRAMIDS
You may already know about pyramids. You might have seen pictures of pyramids in Egypt or Central America. Many ancient cultures made monuments in the shape of pyramids. Pyramids are three-dimensional shapes with one flat bottom and sides that are all triangles. The most common pyramids you see will have bottoms with three and four sides.
CYLINDERS AND CONES
These are some round shapes with either a bottom or top and bottom in the shape of an ellipse or circle. A cylinder is a shape that has ellipses for its top and bottom faces. The side is a smooth surface with no corners. What is a good cylinder shape? Think about fluorescent light bulbs, pipes, or a thermos for cylinders you see every day. You might not see cones every day, but they are out there. Cones look like triangles that have been spun around a central line. They are very sturdy shape and often happen in nature. Go out to a sandbox and pick up a handful of the sand. If you let it out of your hand in a slow stream, it will probably form a cone shape on the ground (round base and pointy top). If you really need an example, think about an ice cream cone. Those cones aren't called cones for nothing.
PRISMS
Prisms are a really cool and used in physics to break white light up into different colors. But what is the actual shape of a prism? The basic idea of a prism is that there are two parallel shapes that can have any number of sides. Those sides are connected by rectangular pieces. The prism used in physics has two faces (top and bottom) that are triangles and three connecting sides that are all rectangles.
MORE 3D SHAPES
There are so many shapes. Let's just say there are way too many to discuss on this introductory page. We have a few favorites that we want to share. A toroid is a donut shape. It's like a tube that has been bent in a complete circle. A helix is a twisting shape. DNA is made in a special helix shape. A corkscrew or spring is a great example of the shape of a helix. A helix can twist in a right-handed or left-handed direction. Let's finish up with a dodecahedron. We just like the name. It's just a simple twelve sided object made up of faces that are in the shape of pentagons (5-sided polygons). Some dice are in the shape of dodecahedrons.
Now we've gone and done it. 3-D. That term stands for three-dimensional. It means an object that has depth. It's an object that you can hold. You are a three-dimensional object. Your computer is a three-dimensional object. Your balls in the toy chest are three-dimensional objects. Everything in the world is three-dimensional. 3-D objects with corners are called polyhedrons in math. It's a word related to polygon (2-D objects). Let's look at a few basic shapes...
SPHERES AND SPHEROIDS
You know that ball you use on the playground? That ball… That perfectly round ball… Is a sphere. Spheres are smooth surfaces with no edges. How do you make a sphere? Take a circle and spin it around its centerline. It is just that simple. Spheroids are round objects that weren't made with perfect circles. A great example of a spheroid is an egg.
PYRAMIDS
You may already know about pyramids. You might have seen pictures of pyramids in Egypt or Central America. Many ancient cultures made monuments in the shape of pyramids. Pyramids are three-dimensional shapes with one flat bottom and sides that are all triangles. The most common pyramids you see will have bottoms with three and four sides.
CYLINDERS AND CONES
These are some round shapes with either a bottom or top and bottom in the shape of an ellipse or circle. A cylinder is a shape that has ellipses for its top and bottom faces. The side is a smooth surface with no corners. What is a good cylinder shape? Think about fluorescent light bulbs, pipes, or a thermos for cylinders you see every day. You might not see cones every day, but they are out there. Cones look like triangles that have been spun around a central line. They are very sturdy shape and often happen in nature. Go out to a sandbox and pick up a handful of the sand. If you let it out of your hand in a slow stream, it will probably form a cone shape on the ground (round base and pointy top). If you really need an example, think about an ice cream cone. Those cones aren't called cones for nothing.
PRISMS
Prisms are a really cool and used in physics to break white light up into different colors. But what is the actual shape of a prism? The basic idea of a prism is that there are two parallel shapes that can have any number of sides. Those sides are connected by rectangular pieces. The prism used in physics has two faces (top and bottom) that are triangles and three connecting sides that are all rectangles.
MORE 3D SHAPES
There are so many shapes. Let's just say there are way too many to discuss on this introductory page. We have a few favorites that we want to share. A toroid is a donut shape. It's like a tube that has been bent in a complete circle. A helix is a twisting shape. DNA is made in a special helix shape. A corkscrew or spring is a great example of the shape of a helix. A helix can twist in a right-handed or left-handed direction. Let's finish up with a dodecahedron. We just like the name. It's just a simple twelve sided object made up of faces that are in the shape of pentagons (5-sided polygons). Some dice are in the shape of dodecahedrons.
2- D Shapes
TWO-DIMENSIONAL SHAPES
What is 2-D? We mean two-dimensional. Aaaand... What is that? A 2-D object is an object with no depth. The closest you will ever come to seeing one is a paper cutout of a shape. But even that paper has a very small thickness, so it is truly 3-D. You might also hear the word polygon. That's another name for a 2D object. We're going to give you a little overview of the polygons you may see in your math travels.
TRIANGLES
A triangle is the generic name for a shape with three (3) sides. There are different types of triangles. You might hear about an equilateral triangle. That is a triangle with all three of its sides at equal lengths. Another type is an isosceles triangle. That a triangle with two (2) sides of equal length and the third side is different.
QUADRILATERALS
A quadrilateral is a shape with four (4) sides. You might hear about squares, rectangles, a rhombus, or a trapezoid. These are all different types of quadrilaterals. Squares have all sides with equal lengths and right angles (90 degree angles). A rhombus is really close to a square, bit it's at a tilt so that there are no right angles. It is like a diamond on its side.
PENTAGONS AND HEXAGONS
Moving up in the number of sides we find pentagons with five (5) sides. Next in the number of sides, we discover hexagons. Hexagons have six (6) sides. You can add even more sides to polygons. There is no limit. We're also doing something different with these pics. You should know that a polygon could be any shape, even all bent and funny. If all of your sides have equal lengths, you are a regular polygon.
ELLIPSES
What happens if you only have one side? Is that possible? Kind of. Ellipses are circular shapes that could be circles or ovals. The border goes all the way around and there are no corners. You could also think about it as a polygon with an infinite amount of sides that all blend together.
OTHER SHAPES
You will find other types of shapes in the math world. Cardioids are like circles with a dimple on one side. A good example of a cardioid is a heart shape. You will also find stars. Typical star shapes are the five-pointed kind. Stars are created by connecting the corners of shapes with more than four sides. The list goes on, but not on this page.
What is 2-D? We mean two-dimensional. Aaaand... What is that? A 2-D object is an object with no depth. The closest you will ever come to seeing one is a paper cutout of a shape. But even that paper has a very small thickness, so it is truly 3-D. You might also hear the word polygon. That's another name for a 2D object. We're going to give you a little overview of the polygons you may see in your math travels.
TRIANGLES
A triangle is the generic name for a shape with three (3) sides. There are different types of triangles. You might hear about an equilateral triangle. That is a triangle with all three of its sides at equal lengths. Another type is an isosceles triangle. That a triangle with two (2) sides of equal length and the third side is different.
QUADRILATERALS
A quadrilateral is a shape with four (4) sides. You might hear about squares, rectangles, a rhombus, or a trapezoid. These are all different types of quadrilaterals. Squares have all sides with equal lengths and right angles (90 degree angles). A rhombus is really close to a square, bit it's at a tilt so that there are no right angles. It is like a diamond on its side.
PENTAGONS AND HEXAGONS
Moving up in the number of sides we find pentagons with five (5) sides. Next in the number of sides, we discover hexagons. Hexagons have six (6) sides. You can add even more sides to polygons. There is no limit. We're also doing something different with these pics. You should know that a polygon could be any shape, even all bent and funny. If all of your sides have equal lengths, you are a regular polygon.
ELLIPSES
What happens if you only have one side? Is that possible? Kind of. Ellipses are circular shapes that could be circles or ovals. The border goes all the way around and there are no corners. You could also think about it as a polygon with an infinite amount of sides that all blend together.
OTHER SHAPES
You will find other types of shapes in the math world. Cardioids are like circles with a dimple on one side. A good example of a cardioid is a heart shape. You will also find stars. Typical star shapes are the five-pointed kind. Stars are created by connecting the corners of shapes with more than four sides. The list goes on, but not on this page.
Pi
Pi
Pi sounds like pie and is equal to about 3.1416. In math, this is the ratio of the circumference of a circle to its diameter. In other words, pi is a number that equals the quotient of the circumference of a circle divided by its diameter. Many people celebrate pi by holding a Pi Day on March 14th or 3/14.
The Greek letter pi represents the number by which the diameter of a circle must be multiplied to obtain the circumference. Pi is an irrational number. That is, it cannot be written as a simple fraction or as an exact decimal with a finite number of decimal places. However, you can increase the number of digits until you reach a number as near to pi as needed. Mathematicians with computers have calculated pi to millions of decimal places.
Pi is used in several mathematical calculations. The circumference of a circle can be found by multiplying the diameter by pi (c = pi X d). The area of a circle is yielded by multiplying pi by the radius squared (A = pi X r-squared). Pi is also used to calculate the area of a circle, and the volume of sphere or a cone.
The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery'[1] and περίμετρος 'perimeter', i.e. 'circumference'.
π is defined as the ratio of a circle's circumference to its diameter:
Circumference = π × diameter
Note that the ratio c/d does not depend on the size of the circle. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference c, preserving the ratio c/d. This fact can also be stated as saying that all circles are similar.
Area of the circle = π × area of the shaded square
Alternatively π can be also defined as the ratio of a circle's area to the area of a square whose side is the radius:
The constant π may be defined in other ways that avoid the concepts of arc length and area, for example, as twice the smallest positive x for which cos(x) = 0.[2] The formulæ below illustrate other (equivalent) definitions.
Numerical value
The numerical value of π truncated to 50 decimal places is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
Pi sounds like pie and is equal to about 3.1416. In math, this is the ratio of the circumference of a circle to its diameter. In other words, pi is a number that equals the quotient of the circumference of a circle divided by its diameter. Many people celebrate pi by holding a Pi Day on March 14th or 3/14.
The Greek letter pi represents the number by which the diameter of a circle must be multiplied to obtain the circumference. Pi is an irrational number. That is, it cannot be written as a simple fraction or as an exact decimal with a finite number of decimal places. However, you can increase the number of digits until you reach a number as near to pi as needed. Mathematicians with computers have calculated pi to millions of decimal places.
Pi is used in several mathematical calculations. The circumference of a circle can be found by multiplying the diameter by pi (c = pi X d). The area of a circle is yielded by multiplying pi by the radius squared (A = pi X r-squared). Pi is also used to calculate the area of a circle, and the volume of sphere or a cone.
The constant is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery'[1] and περίμετρος 'perimeter', i.e. 'circumference'.
π is defined as the ratio of a circle's circumference to its diameter:
Circumference = π × diameter
Note that the ratio c/d does not depend on the size of the circle. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference c, preserving the ratio c/d. This fact can also be stated as saying that all circles are similar.
Area of the circle = π × area of the shaded square
Alternatively π can be also defined as the ratio of a circle's area to the area of a square whose side is the radius:
The constant π may be defined in other ways that avoid the concepts of arc length and area, for example, as twice the smallest positive x for which cos(x) = 0.[2] The formulæ below illustrate other (equivalent) definitions.
Numerical value
The numerical value of π truncated to 50 decimal places is:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
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