page_title: Multiplication
section_title: Computation
hierarchical_section_title: Multiplication / Computation
caption_attribution_description: עברית: צעצוע מכני משנת 1918 לחישובי לוח הכפל
context_page_description: Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product.
  The multiplication of whole numbers may be thought of as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier. Both numbers can be referred to as factors.
  For example, 4 multiplied by 3, often written as and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together:
  Here, 3 and 4 are the factors, and 12 is the product.
  One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3:
  Thus the designation of multiplier and multiplicand does not affect the result of the multiplication.
  The multiplication of integers, rational numbers and real numbers is defined by a systematic generalization of this basic definition.
context_section_description: The common methods for multiplying numbers using pencil and paper require a multiplication table of memorized or consulted products of small numbers (typically any two numbers from 0 to 9), however one method, the peasant multiplication algorithm, does not.
  Multiplying numbers to more than a couple of decimal places by hand is tedious and error prone. Common logarithms were invented to simplify such calculations, since adding logarithms is equivalent to multiplying. The slide rule allowed numbers to be quickly multiplied to about three places of accuracy. Beginning in the early 20th century, mechanical calculators, such as the Marchant, automated multiplication of up to 10 digit numbers. Modern electronic computers and calculators have greatly reduced the need for multiplication by hand.
caption: The Educated Monkey – a tin toy dated 1918, used as a multiplication “calculator”. For example: set the monkey’s feet to 4 and 9, and get the product – 36 – in its hands.
exif: {"Image Make": "LG Electronics", "Image Model": "LG-D855", "Image Orientation": "Horizontal (normal)", "Image YCbCrPositioning": "Centered", "Image ExifOffset": "2182", "GPS GPSLatitudeRef": "N", "GPS GPSLatitude": "[31, 46, 296707/5000]", "GPS GPSLongitudeRef": "E", "GPS GPSLongitude": "[34, 37, 202487/5000]", "GPS GPSAltitudeRef": "0", "GPS GPSAltitude": "35599/1000", "GPS GPSTimeStamp": "[6, 19, 10]", "GPS GPSDate": "2018:06:14", "Image GPSInfo": "4586", "Image Padding": "[]", "EXIF ExposureTime": "1/24", "EXIF FNumber": "12/5", "EXIF ISOSpeedRatings": "100", "EXIF ExifVersion": "0220", "EXIF DateTimeOriginal": "2018:06:14 09:19:21", "EXIF DateTimeDigitized": "2018:06:14 09:19:21", "EXIF ComponentsConfiguration": "YCbCr", "EXIF ExposureBiasValue": "0", "EXIF Flash": "Flash did not fire", "EXIF FocalLength": "397/100", "EXIF SubSecTime": "327331", "EXIF SubSecTimeOriginal": "32", "EXIF SubSecTimeDigitized": "32", "EXIF FlashPixVersion": "0100", "EXIF ColorSpace": "sRGB", "EXIF ExifImageWidth": "4160", "EXIF ExifImageLength": "2340", "Interoperability InteroperabilityVersion": "[48, 49, 48, 48]", "EXIF InteroperabilityOffset": "4568", "EXIF WhiteBalance": "Auto", "EXIF Padding": "[]"}