(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fmadblacktrans.gif&hash=702c52262ea9e51633203bcf8ed56dde347d0d9d)
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fzebrarule.gif&hash=419fb3e492f4e58cc532e1c9b93f2558e7770923)
EYPTIAN COUNTING WITH HEIROGLYPHS
| These are the basic glyphs (symbols) used in Egypt for counting over 4000 years ago:
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)=1
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)=10
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall2.gif&hash=680d30af33cbf8da6eb696df38191dd638b4fac1)=102=100
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall3.gif&hash=3f616be87ef56bab343531215b6cddbf257e1b68)=103
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall4.gif&hash=786e2b761eb128cb5e48fe582e88095185efb713)=104 =10000
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall5.gif&hash=dc2060983c1a8ac8fd14e317150bc1df48795b91)=105
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall6.gif&hash=008e1ef909816a2a0f78bc30e3b5d20bc97203a6)=106
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall7.gif&hash=ef031de5c4eac5e1c53571c85f43057149743565)=107
|
Writing an integer consists of writing the number (from 0 to 9) of the proper symbols to represent the integer. Thus,
3105 = 3*(1000)+100+5 = (https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall2.gif&hash=680d30af33cbf8da6eb696df38191dd638b4fac1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall3.gif&hash=3f616be87ef56bab343531215b6cddbf257e1b68)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall3.gif&hash=3f616be87ef56bab343531215b6cddbf257e1b68)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall3.gif&hash=3f616be87ef56bab343531215b6cddbf257e1b68).
There is also a glyph(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptequalsmall.gif&hash=89fac5c5dd6f9307689157ecee94c2f0dfdd39b2)which can translated as "equals" and a compact way of writing large glyphs, as shown below on the right, for two ways 35:
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52) (https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptequalsmall.gif&hash=89fac5c5dd6f9307689157ecee94c2f0dfdd39b2) (https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol35.gif&hash=c5d0175f22a9dc7cda0c2d4daf1a2bedc0047098).
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fzebrarule.gif&hash=419fb3e492f4e58cc532e1c9b93f2558e7770923)
Addition & Subtraction
in early Egypt
| Addition and subtraction were simple processes using the counting glyphs (http://www.math.buffalo.edu/mad/Ancient-Kmt/mad_ancient_egypt_arith.html#couting%20glyphs). To add two numbers, collect all symbols of similar type and replace a ten of one type by one of the next higher order. For example, adding 35 and 17:
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol35.gif&hash=c5d0175f22a9dc7cda0c2d4daf1a2bedc0047098)
add
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol17.gif&hash=d19c79515055db8836ef92aa72ae5cc3370ff5a5)
| (https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall1.gif&hash=ff859adb2e722fbdbb7736017c5991a96f5cbf52) | (https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptequalsmall.gif&hash=89fac5c5dd6f9307689157ecee94c2f0dfdd39b2)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol52.gif&hash=5b006b65057d8c9f4b36b4ddb5847dd10d80931f) |
Subtraction is a reversal of the process, if necessary replace a higher, so
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol35.gif&hash=c5d0175f22a9dc7cda0c2d4daf1a2bedc0047098)
subtract
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol17.gif&hash=d19c79515055db8836ef92aa72ae5cc3370ff5a5)
| (https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptequalsmall.gif&hash=89fac5c5dd6f9307689157ecee94c2f0dfdd39b2)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol52.gif&hash=5b006b65057d8c9f4b36b4ddb5847dd10d80931f) |
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fzebrarule.gif&hash=419fb3e492f4e58cc532e1c9b93f2558e7770923)
Multiplication and Division
| Multiplication and Division (http://www.math.buffalo.edu/mad/Ancient-Kmt/mad_ancient_egypt_arith.html#division) were also simple processes using the counting glyphs (http://www.math.buffalo.edu/mad/Ancient-Kmt/mad_ancient_egypt_arith.html#couting%20glyphs). To multiply two numbers, all you needed to understand was the double or the half of an integer; i.e., the 2 times table. For example, to multiply 35 by 11. You successively multiply double 35 (and its doubles) at the same time as doing this to 1, until just before you get to 11 in the latter. Below we first do this in present day notation, and then give the translation into glyphs the way the ancients would have done.
| |
35
|
2*
|
70
|
4
|
140
|
8*
|
280
|
1+2+8=11
|
35+70+280=385
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol35.gif&hash=c5d0175f22a9dc7cda0c2d4daf1a2bedc0047098)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymboltwo.gif&hash=06b1f32d585caa0f1bea92a801441058d6ba02d0)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol70.gif&hash=bb87a50967dd73bcfffe02791d3e0f80bd6342c6)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolfour.gif&hash=e298b378d06d01dda90e286fba076d317a77cd5a)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol140.gif&hash=5433bab7605c8e6cf0ba890da10b9c5fe8558f3e)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol8.gif&hash=0c3c38ec7e1ec8c53ba05a22a10a42812c66330c)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol280.gif&hash=5b4359b1f58097a3b8644a632d9657c0c06060be)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbolsmall0.gif&hash=b0a39b70870349f87bcd9802023efd5c1e505bc1)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymboltwo.gif&hash=06b1f32d585caa0f1bea92a801441058d6ba02d0)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol8.gif&hash=0c3c38ec7e1ec8c53ba05a22a10a42812c66330c)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptequalsmall.gif&hash=89fac5c5dd6f9307689157ecee94c2f0dfdd39b2)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol11.gif&hash=8b950abcc67f567a82d85a239add753860d3ac04)
|
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol35.gif&hash=c5d0175f22a9dc7cda0c2d4daf1a2bedc0047098)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol70.gif&hash=bb87a50967dd73bcfffe02791d3e0f80bd6342c6)(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol280.gif&hash=5b4359b1f58097a3b8644a632d9657c0c06060be)
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptequalsmall.gif&hash=89fac5c5dd6f9307689157ecee94c2f0dfdd39b2)
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fegyptsymbol385.gif&hash=fd044cc90854cfc0da117a412a070545d653ec36)
|
Division, up to fractions, is just a reversal of the multiplication process, where it seems the egyptian did not think of, say "divide 1075 by 25." Instead the question was posed as "how many times should25 be added to itself to yield 1075." Specifically, to divide 1075 by 25, again double,1 on the left , 25 on the right until just before you surpass 1075. Unless we have no whole number divisor, some choice of sums on the right equals 1075. Choose the corresponding rows on the left
1
| |
25*
|
2
| |
50*
|
4
| |
100
|
8
| |
200*
|
16
| |
400*
|
32
| |
800*
|
1+2+8+32=43
| |
25+50+200+800=1075
|
What happened when one divided 1079 by 25? We would write 43+4/25. However, the Egyptians did not write the fraction 4/25. Instead, you might see 1/10 + 1/20 + 1/100 or 1/8 + 1/40 + 1/100 (see egyptian fractions (http://www.math.buffalo.edu/mad/Ancient-Kmt/mad_egyptian-fractions.html)).
|
Egyptian Arithmetic - Mathematicians of the Kmtyw Diaspora (http://www.math.buffalo.edu/mad/Ancient-Kmt/mad_ancient_egypt_arith.html)
(https://www.abibitumikasa.com/proxy.php?request=http%3A%2F%2Fwww.math.buffalo.edu%2Fmad%2FAncient-Kmt%2Fzebrarule.gif&hash=419fb3e492f4e58cc532e1c9b93f2558e7770923)
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