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Many instances in the Talmud involve the number 60 as for instances:
Why was the number 60 chosen by the Sages and not 100 or 50 or 10? What is the significance of 60?
Regards,
The concept of nullifying forbidden food when mixed with kosher food 60 times its volume is discussed in the Gemara (Hulin 98a-b), followed by the biblical source for this number: the foreleg of the nazirite's ram.
וְלָקַ֨ח הַכֹּהֵ֜ן אֶת־הַזְּרֹ֣עַ בְּשֵׁלָה֮ מִן־הָאַ֒יִל֒ וְֽחַלַּ֨ת מַצָּ֤ה אַחַת֙ מִן־הַסַּ֔ל וּרְקִ֥יק מַצָּ֖ה אֶחָ֑ד וְנָתַן֙ עַל־כַּפֵּ֣י הַנָּזִ֔יר אַחַ֖ר הִֽתְגַּלְּח֥וֹ אֶת־נִזְרֽוֹ
The priest shall take the cooked foreleg of the ram, one cake of matza out of the basket, and one matza wafer, and shall put them upon the hands of the nazirite, after he has shaved his consecrated head. (Numbers 6:19)
If the foreleg is cooked along with the entire ram, the rest of the ram is still kosher for non-priests to eat. The sages determined that the foreleg is one-sixtieth of a ram, which establishes the general concept:
מאן דאמר בששים סבר בשר ועצמות בהדי בשר ועצמות משערינן והוה ליה בששים מאן דאמר במאה סבר בשר בהדי בשר משערינן והוה ליה במאה
The Gemara now returns to the dispute about whether non-kosher food is nullified in sixty or one hundred times its volume of kosher food, and explains how each opinion is derived from the halakha of the foreleg of the nazirite’s ram. The one who said that non-kosher food is nullified in sixty times its volume of kosher food holds that we assess the ratio of meat and bones of the foreleg to the meat and bones of the rest of the ram, and this is a ratio of one to sixty. (Steinsaltz Talmud)
The other cases of 60 in the Talmud appear together in Brakhot 57b:
אֶחָד מִשִּׁשִּׁים לְגֵיהִנָּם. דְּבַשׁ — אֶחָד מִשִּׁשִּׁים לַמָּן. שַׁבָּת — אֶחָד מִשִּׁשִּׁים לָעוֹלָם הַבָּא. שֵׁינָה — אֶחָד מִשִּׁשִּׁים לַמִּיתָה. חֲלוֹם — אֶחָד מִשִּׁשִּׁים לַנְּבוּאָה.
Our fire is one-sixtieth of the fire of Gehenna; honey is one-sixtieth of manna; Shabbat is one-sixtieth of the World-to-Come; sleep is one-sixtieth of death; and a dream is one-sixtieth of prophecy.
It's logical to assume that these cases (fire, honey, Shabbat, sleep, and dreams) follow the food nullification principle in that they are marginal when compared to a grander model. In fact, this is precisely what the Ben Ish Hai says in his commentary (Ben Yehoyada) on the sugya in Brakhot:
כל שיעורים אלו של אחד מששים דנקיט הכא אינם אחד מששים, אלא ודאי שיש בהם יתרון למאות ואלפים ויתר כל אחד לפי עניינו, והא דנקיט בכולהו אחד מששים, מפני כי בעלמא שיעור הביטולים הוא בששים, שכל דבר אסור יתבטל אחד בששים וחשיב כאלו אינו כלל ועיקר, וכן הענין בא ללמדנו, אש המצוי כה יתבטל כוחו לגבי כח אש גיהנם כמו ביטול האיסור שיתבטל אחד בששים וחשיב זה חלק האחד כאילו אינו, אבל לעולם ודאי כח אש גיהנם הוא עצום ורב וגדול על כח אש המצוי אצלינו יותר ממאה אלפי פעמים וכן הענין בשאר מינים אשר מזכיר פה.
All of these one-sixtieth measurements are not actually one in sixty... but they're all called that because nullification is measured as one-sixtieth. Every forbidden thing is nullified in a volume of sixty, where it is then considered marginal and unessential. The principle teaches us that the power of common fire is nullified with respect to the fire of Gehenna, just like a forbidden thing is nullified in a volume of sixty. This one part is considered as if it did not exist, unlike the power of Gehenna fire, which is eternally vast and mighty. It is more than a hundred thousand times greater than the power of common fire that we have. And the same [metaphorical analogy] is true of the other cases brought down here.
A more mathematical answer shows that working with parts of 60 is actually a very efficient mathematical choice, especially in Babylonean times. To this day we still work with '60', such as in our system of time, the 12 star signs, etc.
To understand why parts of 60 work, you need to understand what a mathematical base is. Each digit raises the base to power based on the position (going right to left). The first digit is the base to the power 0, the second digit is the base to the power 1, etc.
We are used to a base 10 system, so when we add things we count up to nine, before starting anew in a cycle. 11 means '10 and 1'. Most are also familiar with the base 2 system, aka the binary system. You count from zero to one, before 'restarting the clock'. 11 means '2 and 1'.
It is known from ancient texts that in the Babylonian region a base 60 system was used.
This system has some incredible properties in regards to counting without tools, such as an abacus. In a base 10 system, we count on our fingers to ten, and then things start to get difficult. How do you count to 11 on your fingers? In a base 60 system that is much easier, and the Babylonian writings reflect their 'counting on fingers system'.
Take your left hand, and don't count on your thumb. Start counting the sections of your fingers. You have four fingers (if you exclude the thumb), with three parts each, so twelve parts in total. The trick is to count up to twelve by pointing on your 12 finger-parts with your index finger. As soon as you have counted 12, you start anew by counting with your index and middle finger. This allows you to count up to 24. After adding your ring finger and ultimately your pinky, you will have reliably counted up to 60.
The base 60 system also has other big advantages for a world without computers. 60 is easily divided into parts, which is why it is still popular in our time-system, which is also a base-60 system (60 minutes in one hour). If you need to divide it in half, you get 30, which can be divided in quarters to get 15 etc. But most importantly: 60 has a wonderful property woefully lacking in the base-10 system, and that is that it is divisible by 3. The fraction 10/3 is much more difficult to work with than 60/3, because 10/3 is not an integer, whereas 20 is a whole number.
Base 10 can be divided into whole numbers by two factors only, namely, '5' and '2'. Other systems have advantages over base-10, especially in the ancient world that hadn't invented the decimal points yet. Base 30, the smallest base that is divisible by 2, 3, and 5. Base 60 is at the tipping point where it can be counted 'naturally' such as explained above, and is divisible by 2, 3, 4, 5, 6. Higher than sixty doesn't really help improve upon base-60. To get a clean fraction like 1/7 using an analogous representation, you’d have to go up all the way to the basically unusable base 210 system.
A bit more technical: base 10 only has the prime factors '2 x 5', whereas the prime factors of 60 are '2 × 2 × 3 × 5'. More prime factors are a good thing if you need to divide things in parts. Any system that allows you to divide things in clean equal parts is superior to a system to forces you to 'wing it' when you need to divide it in anything other than 2 or 5. As mentioned, you need to go up to 210 to get a qualitative improvement upon 60 with the prime factors '2 × 3 × 5 × 7'. But counting in cycles of 210 is a cognitive strain of biblical proportions.
So, in a world where you count, and most importantly, divide a lot of 'real' things (as opposed to the more abstract things we use mathematics for nowadays), and you don't have the luxury of computers, the base-60 system has a lot of advantages over other systems of counting, mostly because it has a lot of prime factors, i.e. is easy to divide into whole numbers. Base-10 has other advantages, but in the old world of the Babylonian Talmud the base-60 system is superior.
